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Axiom !: The Experience "I" have is solely my own and is constructed by "me" within a context that i can only model using cultural "forms".
Note: The structure of "my" experience i have constructed using the paradigm of a continuum between poles that are indefinitely located in a location that has at least an inner region separated by boundarisation processes from an outer region. It is consequential of the construction process that the outer region has no boundary that cannot be enclosed by another constructed boundary.

Axiom 2: All processes within my experiential continuum are iterative or recursive.

interesting point :D  O0 ;D

Quite so Sir Trifox, quite so; indeed I have spent considerable time meditating on the aforementioned subject, whose delightfully rich consequences I have found on be the whole to quite poignant and enlightening. So much so in fact that, upon later retro-reflection I at once recognised the Whole and Complete Meaning of Life as a direct consequence of those initially-innocuous axioms. Indeed this could be the beginning of a whole new branch of philosophical inquiry, the name for which I cannot presently conceive.

I wonder if our friend Jehovah Jah (Hail King Selassie, Jah Rastafari) would care to expound upon these most solid foundations which he has laid for us.


ps. I notice you are approaching my post count!  O0

One important corollary of Axiom1 is that Everything i construct is necessarily relative to me in "my" model. Hence:
Axiom 3:- i can construct any given number of other reference points within "my" model by iterative processes.

This among other things allows me to rotate {in fact all affine transforms} my model relative to myself so i can gain insights from different vantage points within my model/ experiential continuum. The number of reference points is unbounded. The iterative processes are the basis of "trial and error" within my experiential continuum as i construct the re configured model from my initial assumption of cultural paradigms.

Axiom 4: The "context" in Axiom 1 is not constructed or definable within my model but is perceivable by "me" by a iterative process of negation of all elements within my model.
Basically i can't say what the context is but i can say what it is not by recursive means.

Axiom 5: I stabilise "my" model by an iterative process of "acceptance".

"My" experiential continuum changes with what i "accept" as a basis for the iterative processes of perception and recognition. the cultural forms which i accept from conception are numerous and pervasive and as i alter these my perceptions change as does my experiential continuum. "not altering" then is a nascent notion of acceptance.

Axiom 6: The Set FS is the universal set within which my model/ experiential continuum is defined and has a rule:- all processes on its elements are iterative /recursive and all its elements are determined by iterarive processes. notFS will be the recursive definition of the context in axiom 1. However there is a mapping from notFS onto FS such that FS is a model of notFS.

Axiom 7: Iterative/recursive processes operating on notFS are perceivable.

These processes will be compared with enegetic transfomations within FS.

Axiom 8:All perceived boundaries involve an iterative process or processes.

Definition: Infinfite is unbounded and large Infinitesimal is unbounded and small.

Counting is one of my accepted cultural iterative processes, as is measuring. The perceived boundaries which I count and or measure are the basis of my nascent notion of quantity and support by iterative processes the notion of quality.

The notion of motion within my experiential continuum ls a perception of change in size and relationship to any given boundary or boundaries in a sequential manner. The perception of quickness is that of a sequential boundary change occurring  relative to another sequential boundary change. So for example an object identified by its boundary moves past a boundary in the blink of an eyelid.

Boundaries are what i use to define, identify, count, measure etc and so are inputs into many iterative processes. They also seem to be outputs from the perception process which is an iterative process.

Limits are boundaries beyond which a given process has no effect on what it is operating on.

Orientation is notion perceived relative to boundaries and the sequence they occur in either in motion or in enumeration. "I" have many structures which are involved in the perception process of orientation which i call sensors. It is from the The recursive perception of orientation that i derive the notion of direction.

The recursive processes that "I" engage my sensors in to produce these perceptions are assumed within my description of my experiential continuum.

Points lines and planes are abstract limit surfaces for iterative processes operating on regions whose perceived boundaries are those described in fractal geometry. So in my model I start with a bounded region and iterate to define a plane,line,or point;and as these are limits they do not exist other than as abstractions of a limiting process.
It is also the way hypersurfaces will be identified. This means that surfaces in general or objects will be and already are defined as the end result of some iterative process.

The word algorithm is used to describe a symbolic description of a process. The symbols can be in any language with a proper syntax, that is a sequential way of decoding the actions encoded in the symbols.

I just wanted to briefly discuss the important iterative process of metrication, to highlight how we have always been using fractal patterns practically. In establishing a metric we are doing something very simple and repetetive and recursive in fact. By taking a bounded object as a standard for length we quantified our metric for lenghth. By an iterative process of disection we were able to establish a pattern of marks from this standard length which form the patterns found on all measuring rules or tapes, and this we can continue until the thickness of the marks used become significant. By using finer and finer marks we can produce infinitesimal regions on our standard length. but we can also extend the pattern in an infinite collection of standard lengths which are self similar to every other division within the collection and which have the same structure in orders of 10 throughout. So the metric we use every day is a region within an infinite fractal pattern. It is worth noting that the problem of the thickness of the marks on a boundarised object limiting accuracy has led to a laser beam of a certain wavelength being used as the standard but again the boundary conditions for the bouncing beam are iteratively determined.From this the standard has become the space time constant itself, but again the accuracy of this defintion is subject to an iterative process because of the uncertainty principle. So as i indicated we can see that our standard scientific metric is based upon a fractal pattern which is infinite as well as infinitesimal.

A continuum of points i try to not think in when looking at my experiential continuum. Rather i perceive a continuum of locations, and to reference those locations i have to construct a coordinate system relative to me, the metric i use being culturally determined. The advantgage of this to me is i can explore what is in or at the location in my experiential continuum rather than some abstract points. These apparently are regions with interesting properties which some call spatial. What happens within a region involves some iterative process or processes at many scales of the metric.  These processes are involved with boundaries or within boundaries which are perceived. It is when I try to give the coordinates of the boundary that i find that i am engaged in an infinitesimal process of iteration. This is not strange as the metric itself is defined by an infinitesimal iteration.

There are five fundamatal processes in mathematics: boundarisation, enumeration, operation, mapping, iteration. Of these iteration is fundamental to all the others.
Boundarisation, is a perception led process in which the region of attention is differentiated by a perceived boundary from the entire perception field, such that i am able to perceive one side of the boundary as distinct from the other side.The boundary itself is arrived at by iterative processes within my sensory perception faculty and truncated ones at that. Boundaries are limit surfaces in general and are abstracted from from my experiential continuum by a cultural process of naming and or definition. The iterative process that is involved in boundary formation is highlighted by the simple game of determining if a region is within or on one side of a defined or perceived region called a boundary. As the region gets smaller and closer to the boundary region, the perception of the boundary becomes more and more precise. The boundary may or may not be the limit of this process. If it is not the limit of this process we often define it as such,which tidies things up but hides the iterative foundation of boundaries in general.

The region we use as a boundary may be a symbol of an abstract notion of a boundary; for example a pencil line is often used to represent a boundary called or defined as a line. The pencil line is in fact a region whose boundaries are no more definite than any other region because they to are perceived through an iterative process.

Our perception of boundary is a complex mix of mass, colour, tone, shade and region.

Certain boundaries are named culturally such as line, circle, ellipse etc. others are exemplified such as leaf or petal etc. The abstract concepts of boundary often used in mathematics of a certain type have tended to queer the pitch in favour of non "real" forms which of course have obscured the iterative nature of boundaries and boundarisation.

Enumeration of bounded regions is a cultural process that we are all exposed to. We call it counting. This along with boundarisation forms the basis of our nascent notions of quantity. Quantification is one notion that follows from enumeration,but also order and rank are closely  involved with it. Finally quality arises from the process of enumeration and quantification and rank.

The cultural interation "+1" called counting leads not only to notions of quantity but also of quality. The iteration process is not finishable, therefore not finite. It is not only not finite it is not bounded above. So i am given an unboundable increasing process which contains my notion of infinite. The enumeration of quantities in the form of a metric is the basis of our measurement schemes, and any metric is the result of an iterative process.

Quantification is one of the cultural things that we have done to enable enumeration of things we perceive but which are not bounded. We take a bounded object that is affected by this perceived event. This is used to then quantify these unbounded events. These quantities again are the result of recursive processes. For example the notion of length is quantified by a standard which was at first a proportion of a supposedly fixed distance, but then a standard fixed length of a given material at a given temperature. The uncertainty of this standard led to the use of a certain frequency of electromagnetic radiation to define the length of a meter. Finally the universal constant the speed of light in a vacuum is used to determine the standard length of a meter using a very accurate clock cycle of a given element of caesium. This iterative process will no doubt continue as we find out more refined measures of the quantities we hold as basic.









 

The basic operation or operation pair is addition/subtraction. The basic background to this operation is inclusion or exclusion. From this background I derive also associativity  identity and other subtle operative parameters. It is the boundarisation process that enables me to generalise to inclusiveness and which emphasises the iterative nature of the operation . When I include one more region into a collection of bounded regions I am exhibiting the basic operation of addition and underpinning the iterative process of counting. If I exclude a region from a collection i have to specify a number of things: Is the excluded region already part of the collection- in which case i am performing subtraction; or is it being excluded from counting-in which case i may be performing some set operation on elements within a set, or some algebraic operation on elements within a set and of a certain type.

Multiplication and division are processes wholly derived from addition and subtraction as are integration of a function and diferentiation. These operations then and those derived/uncovered on particular sets are all iterative in form and implementation.

Mathematics has always been a collection of useful calculating tools , methods/procedures and language or notation for representing these aspects of cultural interest and enquiry. So has philosophy. The obfuscation came when the two were combined by western philosophers/ students of the arts. We now have a semi mystical approach to one of the most practical aspects of cultural enquiry namely iterative investigation of boundarised regions in space. The quest for precision and proof for example are philosophical concerns disguised as mathematical. The true nature of mathematics is fun and games! PLAYING WITH REGIONS OF SPACE. Discovering their relationships and working out methods to record the different aspects of those relationships. Geometries despite their seeming abstractness have their root in boundarised regions. The internal properties of boundaries within these regions have occupied diferent cultures and schools of students for those cultures own specific drivers. But the operations on and within boundarised regions have tended to be pushed outside the general conception of mathematics only slowly being identified and included over time in many guises. So addition /subtraction is not questioned, but at one time not now, translation rotation reflection were. Certain properties like symmetry and self similarity cannot be adequately defined without these operations. we have no signs for these operations yet we have marks for stages within an iterative process (addition, counting) which have been given a status not connected to the process itself, a status called number. The numerals are given philosophical status by tradition for example the pythagorean school gave them a whole other mystical aspect, but the iterative operation is overlooked as are/ were many other operations on boundarised regions. When Descartes in particular mixed philosophy with the methods and tools of algebra he created the basis of modern mathematical requirement for rigour.x² +y²=1 the equation for a circle is only that on the region [-1,1] on the x axis, outside of that we have not defined the operations to be performed on the mapping. Whatever those operations are they result in locatons off the plane x,y. The operator i such that i² = -1 is not a number except in a philosophical sense. In this same sense the operators and are numbers, However they are not numerals as they do not mark a stage in the iterative process of counting.

~	1	2	3	4	.......
1	1/1	2/1	3/1	4/1
2	1/2	2/2	3/2	4/2
.
.

By definition every possible fraction can be written in an infinite iterative table of this form. We can devise a sequence which enumerates every fraction in this infinite iterative process. This means that the fractions are countable but of course we are in an infinite process. A little thought convinces that all the fractions which are powers of ten in the denominator are also within this table. Therefore they by syllogistic logic are countable. Since these fractions form the basis of our decimal system it follows that every number representable using these fractions will also be countable. It seems therefore that Cantors description of uncountable infinities may be one of form and not of substance. If we engage in this infinite process we will encounter what we might call infinite fractions. Pi for example or e. Their are infinite fractions which in a limit process equal a simple fraction ¹/₃ is an example. The infinite fraction is ^333../_1000... Instead of countability being the criteria we should look at self similarity, and in particular the ability to match or self reference. So for example every fraction in the table could be matched against a fraction with a power of 10 numerator thus illustrating self similarity, magnification, etc.The problem of assuming that we can list every number in some categorisation scheme and that assumption being proved false is in fact a non problem as indicated above. The problems lie in a lack of rigour around the notion number and the notion of proof, two notions we could possibly do without in favour of numerals and convincing evidence.

What ontological attributes does this "I" have?

FWIW,

db

Axioms 1,3 through 5 are the fundamental ontological attributes for "i" for this system.  ;)

Certain things are undefinable in axiom one. So axiom one is merely a convenient starting point for any ontology. Ontological is to me simply explained by " on to logical". We move from an undefined experiential conglomeration on to a logical sytematic arrangement of experiences" This is in part the unavoidable constructionism that is referred to in axiom 1. More linkable to mapping directly are the inate patterns that we recursively perceive. So for example "straight", "curved" , "crossing" , "angle" , "shape", "boundary" etc are all experiences which to describe we have to exemplify,because we inately perceive these instinctively, but not instantly. Recursion or iteration is necessary to make these perceptions distinct.
Notions like orientation cannot be defined without these inate forms. In this regard "fixed" is an inate iterative experience vital for the concept.
Mapping is a relationship notion with many exemplars, not the least of which is symbolic notation or language itself. For example a word can be related to an object by pointing and uttering the sound or pointing to a symbolic representation of a sound or pointing to an object and the symbolic representation. This by the way is magic in the oldest sense and writing is and always has been the casting of spells as has the spoken narrative or injunctive.
A little thought reveals that our whole experiential continuum is a mapping of sensory data to culturally valued sensory data in culturally valued arrangements or forms for the most part, although we exclude many experiences in this pocess or rather ignore. Those who do not ignore every experience they are culturally encouraged to  may be hailed as a genius or a mad person depending on the utility of what they attempt to map into their society's collective unconscious ie their language. Mathematics borrows from this cultural mix such mapping as mathematicians find useful to illuminate or elucidate the relationships they are exploring whether they be quantities, numerals, permutations,series and sequences etc. Descartes ( following al hourisin) for example perceived a mapping between geometry and algebra for example. This is a descriptive mapping which facilitates certain methods of solution for geometric problems and vice versa algebraic ones. What is often laid to one side is the recursive nature of mapping. This is not to say that this is not pragmatically realised, but rather to say that it is not recognised for what it is: Iteration.

Mapping has an inherent associated process and that is scaling. I have often remarked on the practical number systems that are in use which range from 1,2 many to our modern 1,2,3......... iinfinity. These differences result from scale perceptions. It is the scale perceptions that trigger a switch from counting individually to counting in multiples to eventualy measuring using a quantifier. Each situation is a mapping response to the raw sensory data, and in and of itself reduces detail as i go up the scale.

   i digress here to comment on the meaning of infinite in kinetic situations. In a counting or systematic situation infinite means i can continue forever, but when i think of the term infinite speed or infinite revolutions per second, the term seems meaningless. However we have a scale perception for this situation called 'instantaneous'. Instantaneity is the limit of any time perceptive schema. I do not give time an instantiation in my model . I do however refer to sequence and movement. That I notice and diferentiate periodic movement is an inate pattern perceiving function that i take as part of axiom one along with my shape perceiving function. Instantaneity is equivalent to an eternal 'now' in some philosophical descriptions. However to me instantaneity represents the limit of my ability to process all the kinetic data in and around me. For a computer processor instantaneity would be pushed back much in the same way that high speed film pushes back instant actions for my visual and auditory ability.

Thus scale is an inexorable background quality or attribute of mapping.

Axiom 6 has an interessting corollary. Energy and motion are by it recursive or iterative processes. This leads on reflection to the notion that a set wide iterative process may be a hypothesis worth making with regard to the notions of energy and motion in FS. This set wide process will only be worth making if the energy and motion laws that Einstein derived can be shown to be consistent by every measure with the axioms of the set FS. If this can be done then starting with a suitable fractal rule if recursive processes can be shown to generate einstein like motions and energy equivalents then a convincing case may be made for the recursive action of space. If space has this recursive action it may then be possible to relate each iteration to a notion of sequential statuses which may be similar to the notion of "time" in modern physics. I will be more rigorous in a following post, but essentially the elements of the set FS are many and varied but for it to be useful FS must explain the nature of space or rather have an equivalent definition of space to what is in notFS.

IN being rigorous the notions of recursive or iteration need to be explored and defined. The notion of iteration is the deja vu notion. The latin word iterum reflects the cry of a joyous child "again!" so "I" instinctively know the initial conditions and the process i followed to produce the outcome/ product. "Again" however includes a modified set of initial conditions, modified by the first outcome of the first process. So "I" instinctively sequence the initial states, process, and outcomes. Sequencing in this way is ordering, ranking and eventually arranging by some inate cultural or idiosyncratic schema. The schena itself is influenced by cultural norms and boundary conditions.
My definition of iteration is therefore an abstract from this notion and itself will inflence the notion.

Recursion is a mathematical conception of cyclng or running back on 'itself' to go through 'itself' again which influenced the much more general linguistic notion of recursion found in sanskrit grammar formulations and subsequent grammar formulations. The lingustic form arises as an artefact of attempting to define precisely without an inate understanding that language is a human enterprise that models rather than gives magical control over. The power of language, the alphabets the rules of spell casting etc are all intertwined at this initial phase. Later philosopher magicians grammarians were more able to accept the notion of self evident, that is one must experience it oneself before being able to have a referrent for a symbol or word or group of words acting as a description. I am put in mind of the wonderful creation of artistic  monstrosities which arose from attempted linguistic definitions of what we refer to as a giraffe.

The simpler notion of "again" lies within the more elaborate "running back", as formally running back carries the notion of an initial place to which one is returning from a place at which one turned back. So this place at which one turns back is not an end place so implies a continous action requiring one to race back to the start. But at the start the run back instruction is applied again propelling one toward the turning back place. This conception is so poorly defined that it begs the question what is it one is doing? For this reason alone recursion is presented as an abstruse difficult subject, when in fact it is just a poorly defined one.

Mathematical definitions building on this foundation suffer the same confusing trait. My definition of recursion will therefore be formally equivalent to my definition of iteratiion so that the two notions will be replaced by a single consistent one.

A proceedure is a sequence of actions that act either at the same position in the sequence or in following positions in the sequence or both.

A sequence is an inate organisational structure that i have which is appreciable by me through boundarisation and movement relative to boundaries and is one of a set of inate perceptions including shape angle orientation. extension etc. A sequence is generalised by me to include symbolic arrangements and through synesthesia to include sound patterns and touch and other sense sensation. My very memory structure of these sensations is a contributory factor to my perception of sequence. Culturally sequencing is intertwined with concepts called "time" and though useful and powerful these are not necessary or presuppositionally neutral, neither is "time" inate.
 a sequence may be as long as i like! however some sequences i may use may be truncations of infinite sequences.

Actions or interactions are perceivable changes in a bounded region or at a boundary or both as another perceivable change occurs in a referrence bounded region or boundary in contact in some way with the former.

An Iteration is then a bounded region acted upon by a finite or truncated procedure (the sequence is finite or truncated) whose outcome is then the bounded region that the procedure acts on again.

Axiom 6 requires some careful handling. Axioms 1 - 5 lay out an underpinning framework for 6 but do not define a set or set notation. This is in fact assumed to be the standard mathematical definition and usage. However, the axiom itself is attempting to draw together axioms 1 -5 under a mathematical notation system. Thus axiom 6 is a tautology expressing a symbolic representation (set FS with rule ) of axioms 1 - 5. Tautologies like this are indicative of the iterative nature of my consciousness, and the question arises if the generalised notion of iteration does not preclude me from coming to any other description.

The definition of iteration is clearly to specific to explain everyday usage as I recognise iteration not only on bounded regions but also on values and symbols. Turings machine for example is a symbolic iteration, and Newtons iteration is a value iteration. Then there are the iterations which can be seen in design modification, Editing, scientific inductive reasoning, the scientific method itself acting on a group behavioural process; cybernetic and feedback systems etc. Therefore for FS to represent axioms 1 -5 the rule of iteration must take a more general form which can not rigorously be defined and is subjective to my appreciation of iteration. This is not a new problem. I mention it to highlight the fact that my so called knowledge of set FS is more likely to be a knowledge of a partial or restricted subset of FS or subsets of FS which may or may not be cofactors of one another.

Establishing that the definition of the rule on FS is a weaker form of the definition of iteration clarifies a consistent activity of Mathematicians: They love to focus on specific simple cases which can be well defined or rigorously defined and generalise from those special cases. As they generalise they explore the rules which change to accommodate familiar operations and they define rules to enhance this familiarity. For example the iteration we call counting (addition /subtraction) is used as the design for the union operation in set theory. Therefore the iteration property is inate within this set operation, but more than that the mathematicians have created an iteration of the counting iteration! This is a fundamental example of symbolic iteration where the + has by a creative process been iterated to a U. This begs the question already posed: are iterations endemic to our understanding and perception because we are inately fractal? By fractal i mean the result of an iteration process. Axiom 2 is this systems answer to that question.

Because of the definition of iteration it is more likely that the product of an iteration will have a property called self similarity. Consequently we can use self similarity when it is defined as a measure of how well defined and rigorous the definition of iteration used to produce an object or result is.

Mathematicians study the workings of operators, their larger counterparts algorithms and their iterations. Axiom 2 declares that all results in mathematics are the products of iterations and thus fractals, by my definition. Therefore all mathematics will have the characteristics of fractal geometries and or fractal symbolic logics of values and Turing machines on symbols, to identify a few familiar lines of thought. Take for example the familiar 1 + 1, this is in fact a useful stage in the iteration of counting, or it is a symbolic logic statement. A Turing procedure is at the other end of specification and needs to be precisely specified.Operation of the procedure is fully determined by a finite set of elementary instructions such as "in state 42, if the symbol seen is 0, write a 1; if the symbol seen is 1, shift to the right, and change into state 17; in state 17, if the symbol seen is 0, write a 1 and change to state 6;"

The point of note is that these are only stages in the iterations which can continue for as many times as we need or wish or is determined by some other iteration rule. Fractals are the stuff of my conscious thought, and i as a mathematician specify the area of interest i wish to play with.

The concept of continuous and discrete needs to be viewed in the light of iteration. Graph construction for example is an iterative process and in fact produces discrete locations through which an abstract locus is deemed to move.The continuity therefore is imported into the graph by a property of the abstract locus

I have already remarked on multiplication and division as being repeated addition and subtraction of a fixed value. I will restate this as a form of truncated counting iteration. So the 5 times table is the iteration "+5". All counting iterations are infinite but i truncate them where they are of use to me. ``i can represent this iteration procedure and the truncation by using the notion of multiples or divisions. As a mathematician it is second nature to use a bit of symbolic logic and indeed most of my thinking is cluttered with these symbolic logic statements. What they refer to often i do not know or understand unless i can relate to a learning experience of doing the logical procedure. But even then the playfulness of the procedure is lost under memories of grimacing pedagogues whose scorn or whit drove home the rote learning of the task and not the thrill of exploration.

So 5 X 5 is immediately recognisable but not understood as a truncated iteration procedure expressed in symbolic logic. 5 : 5 curiously is not so well recognised because for me division and repeated subtraction were an unlinked notion until well into my 20's.
 Symbolic logic and the rich history of infighting among mathematicians is also to blame for competing and confusing symbology .

In all counting iterations the space that is being iterated is formally included as 0. For this we have to thank indian mysticism as the greek mystics apparently had no appreciation of the void. A simple but powerful description of the counting iteration is the description of bean counters. A space would be cleared to count the beans and then the iteration begins. The procedure modifies the space by adding 1 bean. This modified space then becomes the basis for the next iteration which is again modified by adding 1 bean. This new modified space is again modified by the iteration through adding 1 bean. These iterations continue ubtil all beans have been added to this modified space.

This procedure links names to each modification of the space, and each modified space is the referent or the value of the name. It takes a while for the name and the referent to become so conditionally linked that the name has value. The symbol of the name which i call a numeral then has a value attached to it. Modern numerals we can trace back to arabic influence but of course there are many earlier name and numeral links. The history of names is linked with a deep mysticism which i call number and numerology, but the symbolic logic of modern maths requires only the numerals. However i acknowledge that the mysticism of former times was and is still a driving motive for playing with the relationships between the values or referents. At the same time the mystics obscured the iterative nature of counting by downplaying or ignoring its fundamental , crucial role in the world around me. It is not a criticism to call iteration boring and so likely to be ignored except by a few extremely ocd monks or ascetics!

A snobbishness arose among mathematicians about elegance and style, which translated into a conceited anachronistic approach to calculators and computers. In addition the reverencing of Euclid in the west in particular led to a false notion of proof. I have to note that this is not a universal feeling among mathematicians but was particular to western european mathematics. Eastern maths has always welcomed any tool that aided the "number crunching".  It is fair to say that if i as a  mathematician pure or otherwise  ignore the tool of computing i am jeopardising the relevance of what i have to say because i can only do a few iterations of my elegant procedures.

The definition of iteration then is strongly represented in counting iterations, and anything directly based on counting iterations will be highly fractal.

Axiom 1 holds together some fairly hard to define but necessary and unavoidable notions. `My' abilities or attributes are only hinted at but implied. The notion of quantity is inate in this axiom as is the notion that i have this ability to quantify. Axiom 1 states explicitly that i can construct an experiential continuum but not in isolation, and that i can model notFS. These abilities are not to be tgnored, for counting iterations are not the basis of my appreciation of quantity. I continually appreciate quantity without being trained to name it in the form of a stage in a counting iteration. Today we have digitising devices that can sense the environment as a signal in myriads of sensors ,and the output from each individual sensor is almost instantaneously given a digital value or a count by a device or circuitry that converts the sensors output into a digital value. What i am alluding to by this is that i have within my perceptual function all the information and more which i am tediously naming and defining in the study of maths. Typically i learnt number bonds and numeral links up to the value of 12, but i could just as easily learnt them up to higher values. Some autistic individuals demonstrate this ability quite well. The thing to note is the sheer power of the iteration that is taking place in the perceptive faculty, and the modern sensor systems allied to the computing platforms give testimony to that.

The perception of boundary is a synesthesia of several sensors of which vision is the dominant one. It is not clear what boundary can be perceived by the ear say which does not immediately elicit a stored image of the same or similar. Similarly the sense of touch is so local that image perception is key to it being in any way indicative of boundary. Orientation extensibility etc all make more powerful sense in the visual sensors but nevertheless have contributory components in the other sensors. Shutting my eyes for example and trying to explore the orientation of my teeth in my mouth by touch of my tongue or finger reveal the intricate link to the image in my visual faculty. Therefore boundary is a perceptual iteration result with many sub systems to the visual contributing. Consequently the abstraction i call a line is an overlay of an image onto a perception of boundary. This image can be drawn from visual experiences of say silk or spider web or even a sharp pencil line. To abstract it further to nothing but extensibility is a faux pas that leads to certain inconsistencies.

The notion of straight and parallel derive from boundary segments that have certain inate properties. These properties are explored by euclid quite well and do not require abstract infinite lines. The notion of circle and spiral and oval derive from natural observations, but for me to observe them i must have some inate sense of curve, and an iteration process for that sense makes good sense of the recognition or perception procedure.

The context referred to in axiom ! is undefined and undefinable which is why i define my experiential continuum and formalise it in the set FS in axiom 6. However much of my inate abilities and functions which are in the set notFS  are gradually by iteration processes being found to be mapped / modeled in the set FS. The vision system within my symbiotic microbial colonic system has a counterpart in the ccd and cmos sensor systems used within cameras and electronic visidn systems. A study of the biological vision system : the eye the optic nerve and the visual cortex reveals that the retina is modeled by the pixel system the analogue/digital converter is modeled by modifier cels just behind the retina and the visual cortex models the digital/signal processors. Their is also a clear regular pattern of rod and cone arrangement in the retina which provides a grid like arrangement. This grid like arrangement is mimiced in the arrangement of decoding neurons in the visual cortex. The eye and the visual cortex developmentally (by iteration) are extensions of each other. These regular arrangements of cells, crystal lattices,packing of small objects, molecules etc are being studied under the heading of self assembling structures. Suffice it tosay that my inate sense of shape angle line boundary and orientation can be found in the patterns that this vision system is able to respond to in conjunction with the other sense sub systems.So for example a boundary arises when an arrangement of rods and or cones fires off at a particular action potential within a region on the retina and a different action potential on the "other " side of that region. The arrangement of rods and cones are the inate shape or angle or line which the processing cortex uses to engage in the perception iteration. Somewhere along that set of iterations i am able to make a connection with a stored model that gives rise to the recognition response. The iteration then proceeds but a "higher" level iteration now dominates and uses the recognition iteration in a verification iteration. IF the verification is not found then i may begin the recognition iteration again from a different perspective, whether that be a different angle or a closer(magnified) look until verification is achieved. This process is the basis of curiosity, and i may never achieve verification so i may always be curious about some experience. At another level of functioning this particular non verification may be used to detect a whole class of similar situations. 

Axiom 2 needs to be further explored as to the all inclusive nature of its statement. Clearly some iteration will have to be defined as null iterations if it remains in its present form. However how to distinguish between  a null iteration and a fixed result iteration may be a valuable thing to explore with regard to iterations that transfer energy into a fixed region, gravity for example.

I am setting down a few notes about the conception of time and space common in the west. Axiom 2 declares that these are fractals. It is important to distinguish time because this is a fractal used as a foundational scale like length but is based on movement in space. The fractals that i am looking at are therefore space and movement in space.

Like all conceptions there is an inate sense of the fractal we call time, but what i am sensing is the iteration process or procedure. Axiom 3 declares that i am able to have multiple orientations on this inate sense and my chief sense is how personal to me the iterations are. Then as i look out or sense other bounded regions i can experience a different iteration process happening there within or including the boundary region. I can also find similar iteration procedures going on at a larger scale and at larger scales than that. These larger scale iteration procedures imply that the iterations processes within smaller scale boundaries are not isolated from each other by the perceived boundaries but may in fact be entrained to the larger scale ones. This causative relationship is perceived but as "chaotic" or complex. To this list of descriptors i may now add fractal. But it is not just a descriptor it is a fundamental reorganisation of any notions of cause and effect i have formerly introduced.

Axiom 9 therefore i sketch out here as: large scale and small scale iteration procedures in FS are fractally entrained at any scale size i wish to examine.

When i take a large scale iterative process such as a solar cycle as a standard or the rotaton of the earth about its axis, i can then subdivide it into smaller and smaller segments and use it as a measure or metric. Measures or metrics are some of the most obvious fractals that i can create or design in FS, but as such they are abstracts. Each iteration process i use as standard has to include the sub iterations within its "orbit" to imply any useful iteration link. So to become so abstract that a metric is applied "outside" of its defining iteration procedure is likely to lead to problems of scale.

When i experience a fractal zoom it reminds me so strongly of the differences in structure which scale changes reveal and therefore it is a wisdom to me not to generalise in an assumption of a "smooth" continuous development beyond a certain iteration procedures  defining region of operation. Rather i should expect discontinuity and discrete regionalised developments. So for example the quantum physics and classical physics are fractally entrained by axiom 9 so they will operate on each other,but there will not be a smooth continuous link between them. However i can approach iteratively close to the "boundary" between them in a wada "point" sense. I could refer to this as asymptopic, but i am not assuming a smooth continuous progression.
 
So using the movement of space in space i have traditionally accepted an abstract conception of iteration procedures and caled this "time". What i have perceived within the iterations is movement, and in movement the speed or rate of movement. So using my own visual sensors and the inate notions of sequence position/location and my own internal iteration procedures i am able to develop a sense of speed or rate. Even within my own body if i take my heart beat as a standard i can develop a rate assessment of say how fast i am breathing. This is a useful example as it reveals that many iterative internal procedures are combined in developing this sense of rate of breathing besides heart beat, and again axiom 3 leads me to expect this.

Space and movement of space in space under an iterative procedure is what i will look at next post.

The following are candidates for Axioms of set FS but i have just considered them so i have yet to assess them.

Inertia, Equilibrium, Syntax, Parsing and Equivalence.

By Equivalence i mean that things `are not the same despite the objects being seemingly identical. Every thing is unique but some things are apparently the same, that is perceived similarity, or apparently identical, that is perceived congruency. The class of sets of "things" for which these relationships are a good description i am calling equivalence, and the specified sets within that class i am calling equivalences. The specification of a set in Equivalence is those aspects of a "thing" which are perceivable  as the same in all elements of that set. These sets are in general containing similar elements. A set  which is consisting of similar elememts which upon further iterative investigation are found to be the same for any arbitrary new rule imposed on the set is defined as a set of identical elements and called a congruence.

For a given set in Equivalence we can form  ratios of its elements specific properties such that for a property p in the set the ratio of p for element i [ p_i] to the same property p for element j [p_j]  is p_i  :  p_j  or alternatively . This ratio wherever it makes sense for elements i and j will be a constant value in the sense that the ratio will always be reducible to the smallest numerals by "dividing out" common multiples.

I can establish a system of ratios based on the same property for many elements in a specific equivalence and these allow me to represent  an operation or an action or a relation that applies to many elements at once. This also enables me to represent the many levels of a fractal system by one convenient ratio.

Between specific equivalences there may also be a property that allows a ratio to be formed. Such ratios between equivalences are in the Equivalence class and may represent relationships of one equivalence acting on another. The action of an equivalence on another may be indicated and a property linking the two equivalences be implied by the ratio between the equivalences being constant.

Thus the iterative process of enquiry may be characterised by the forming of Equivalences, the forming of ratios around a common property, and the seeking of properties to link found constant ratios.

Many things i take for granted are equivalences. For example mass is an equivalence. My mass is the equivalent mass of water at a certain temperature that has the same moment on a balance scale as i do. The specific property is the moment on a balance scale. By this property many equivalences can be put into ratio with each other. However these properties may not specify the set of the participating equivalences, rather they define a new set a new equivalence.

Before i continue i want to sketch out a possible universal iteration procedure. I am thinking of a relative vortex for each individual. So the entire experiential continuum i have constructed is based on a procedural vortex relative to myself. Each iteration applying the vortex results in motion within my experiential continuum. At each fractal level fractal entrainment across the boundary generate motion on the other side of the boundary and either side of the boundary. The vortex procedure moves a region at each iteration to a new position, and regions within regions to new positions within the regions. The universal application does not imply uniformity, it implies fractality at all scales, which is to say that the product of the iteration under the vortex procedure is a fractal with infinite levels, and these products would be vortex motions at all scales. Wile this may appear to one observer to be chaotic to myself it would appear fractal, and would generate a search for the self similarity ratios the boundaries of regionalisation and the evidence of fractal entrainment. As this vortex procedure is universal, all motion that results will be voticular to scale. Thus i would expect to find that all forms of motion from seemingly straight line motion to hyperbolic parabolic elliptic circular, cardioidal  and spiral and even brownian would be apparent in its region of operation, which is universal and thus at all scales.

Since we use elliptical and circular motion to define periodicity i would expect periodical forms of motion to be linked to the iteration cycle of the universal vortex iteration in some way.

I would also expect brownian motion to be linked to vorticular motion and fractal entrainment both ways across a boundary with a wada basin condition.

Whatever descriptions we have of vorticular procedures should have this fractal nature if this is a universal iteration, and boundary conditions will need to be generalised to reflect the wada nature of all boundaries in a fractal.

http://www.youtube.com/watch?v=hPgq4AbLKyU&feature=PlayList&p=671AE3C76FC085B5&index=18

The vortex procedure in the set FS will have properties demonstrated above. Thus regions around the vortex will be moved in a coherent way and at different scales. Within the layer generated at each scale by fractal entrainment, regions that are in sympathetic vibratory lock , that is resonance move as a whole with gradual sheer representing growth. Those regions not in resonance move in a way that dissipates them rather like evaporation. Evaporation and condensation can be expected within the layers and across the layer boundaries. Within the layers vortex motion will be evident again fractally entrained by the vortex procedure driving the layer system.

The coherent motion generated by the vortex procedure with these stationary boundary systems and a smooth and flexible boundary condition (at least on a normal everyday scale) exists everywhere even in the twist of the vortex. Only the sheer that cuts spirally down the eye wall of the vortex induces chaotic motion always and spirally on the wall of the eye, Elsewhere resonance induces coherent modes of motion some conformal some sheer, some growth some contraction. Dissonance generates evaporative destructive motion on a region assonance and resonance a conformal motion on a  region and euphony a condensing motion on a region. Notice this coherency in the following

http://www.youtube.com/watch?v=4_72HbbiG6A&feature=related   

The coherency demonstrated here under these conditions allows me to posit that the coherent light produced in a laser or a coherent maser are evidence of vorticular  procedures within the generating crystals under the influence of fractal entrainment by a changing electromagnetic field which itself is vorticular in operation.

With this extreme level of coherency the wada basins for the vorticular motions should have a definite axial shape which demonstrates concentration of the motion along that direction. Also the boundary of a region as a wada basin may be significant in the propogation of fractal entrainment across a boundary at any scale. It may be that all boundaries are wada basins in some sense.

http:"//video.google.com/videoplay?docid=4532767129040787318&ei=NixUSveREYqUqQL1gMGqDw&hl=en#QL1gMGqDw&hl=en#

This tendentious video nevertheless illustrates the conceptual basics but much has to be sifted to get the insight. {copy and paste in the address bar removing the "}

The navier stokes equations for fluid flow of compressible and non compressible fluids have vortex solutions that describe the propogation of vortices in interrupted flows, but the point is not the flow of vortices but the vortex procedure which they describe. This procedure is one model for the universal procedure  in setFS and helps to understand the propogation of wave forms as vortices in this space. For example a vortex or vortex formation can establish a sinusoidal type vibration that is propogated radially or in a flow.

http://www.youtube.com/watch?v=--cPjyxjMdY&NR=1


http://www.youtube.com/watch?v=_AJgEa2dbJU&NR=1

Complicated standing forms can be produced from vortex motion at specific resonances to the medium


http://www.climateaudit.org/?p=466

The calculated solution and the photographed image show not only compliance but other effects due to non resonance.

http://www.youtube.com/watch?v=5_drmTOe424


 

I want such a string-device-thing  :D

An interesting theory of the fractal description of notFS. 


http://www.euclideanrelativity.com/idea/index.asp

So to follow on from a vortex process operating in an iterative way in set FS leads me to re emphasise newtons laws of motion in FS. A body continues in a state of vorticular motion iteration by iteration unless impressed upon by a force. A body impressed upon by a force changes its motion in proportion to the force and along the vorticular path of the force. And finally the impressed or drawn force is opposed by an equal and opposite force acting on the bodies involved  .

Newton accepted the states of motion and rest, but in set FS the iteration of the vortex is the source of all motion by fractal entrainment. A body is only at rest to an observer with the same vorticular motion, and i will discuss this more when i consider equilibrium and inertia.

A body at rest is in a state of force equilibrium which is to say that all forces acting on the body cancel out. Only when this condition fails does a net force impress upon or draw upon a body in a "right" direction in the newtonian sense. This right direction is not a straight line but a perpendicular direction to the place of contact of the force. In a vortex field this approximates to a straight line as a first order approximation, but it is more accurately a logarithmic motion.

That iteration is the fractal entrainment for motion will be a basic axiom of the set FS and will be a development of axiom 2.

i want to note 2 things also for further development.

The cultural iteration '+1' the counting or numbering, that is number naming iteration is based on a very 'natural' process of aggregation as well as a human process of collecting, sequencing, sorting and aggregating. Subtraction or the '-1' iteration is the logical opposite and is based on disaggregation. Natura/Yh~h however provides another formative iterative experience that of splitting into parts. This is not just a biological process but also a physical process of disintegration, found when say a plate drops and breaks into pieces or a rock is shattered into smithereens. It is important to note the wide generality of these 2 processes on which our cultural iterations of addition and fractions  are based. The notion of proportion i have discussed under equivalence in a previous post. it would appear that disaggregation is a similar process to disintegration but at a different scale. The scale however is a psychologically determined unit, which is to say it is a perception determined unity. This of course is just another way of describing the iterative processes involved in the perception process discussed in an earlier post regarding axiom 1 .

Our notions of relativity are inherently based on these iterative perception processes.

The second important point derives from studying the formulae and programmes to produce the mandelbrot set and i will see if it extends to all fractals: a simple aggregation rule is necessary and a simple splitting process is required. To these two requirements a simple iterative force is added to drive the two procedures repeatedly- now i am thinking of waves on a sandy sea shore! By extension all the processes of erosion. In the sea i can visualise a process where single celled creatures are clumped together and dissolved or pulled apart periodically, that is iteratively. When i look for these three things i find them abundantly around me, thus i expect to find real life examples of fractal sculpting. Fractal sculpting is an analogy i have been developing in order to understand what the programmes are doing and how.

So now i am thinking of limestone caves with stalactites and stalagmites as fractals and showing fractal sculpting and structure.The periodic flow of groundwater swelled by seasonal rain, disintegrates the limestone by dissolving the calcium carbonate- that is splitting at the molecular level. Then evaporation aggregates the calcium carbonate at the molecular level causing precipitation of the calcium  carbonate. This means that stalactites and stalagmites are the escape ejecta of the fractal which will be in the cave roof structure. So like rendering can add ornamentation to a fractal by visualising the escape trajectories, or escape speeds stalactites and stalagmites do the same thing.

Another fruitful area for search will be dunes, and wind erosion sites.

It should be possible to use a 2 or 3 variable function to describe the mapping of condensation or crystallization of  molecules in a suspension medium onto a substrate and then through julia iteration to build the shape of the crystal or condensate. Julia iteration is particularly apt for this type of model as a linear function is always a good first approximation.

Equilibrium http://en.wikipedia.org/wiki/Mechanical_equilibrium (http://en.wikipedia.org/wiki/Mechanical_equilibrium) and inertia http://en.wikipedia.org/wiki/Inertia (http://en.wikipedia.org/wiki/Inertia) are 2 sides of the same coin in set FS.
The first thing to acknowledge is that Newton's Laws of motion are the first complete axiomatic field theory. The axioms deal with the elements of motion and force.However the notions of motion and rest are not explored directly in the axioms. Newton was keenly aware of relative frames of reference but still accepted that the true frame of reference would be where his god lived. This he thought of as absolute reference frame in which all measurement and force would have its "true" value.

By stripping down the system to these few axioms, Newton followed euclid in his classical organisational approach to the study of the forcefield and its effect on an object. But newton and his contemporaries had to reduce an object to  point mass to enable this classical euclidean arrangement to utilise as much of euclidean geometry as possible.Mass therefore lost its significance in the pure axiomatic theory to be interpreted as momentum and inertia. Motion also was not defined or explored but assumed and assumed to be modifiable by an invisible hand pushing or pulling through bodily contact.

Newtons third axiom deals with mechanical equilibrium at the points of contact and this was to be the basis of his notion of rest as others derived it. Newton himself saw rest or uniform motion as special solutions to equations making them equivalences in some way that would not be explicit until Einstein's axioms of relativity and relativistic frames of reference.
In setFS motion is the result of iteration and is a vorticular displacement at every scale by fractal entrainment. The fundamental attributes of  Fractal space are axial orientation, axial extension and axial rotation, which is to say spinning around an axis. From these three i derive a special axis for orientational rotation which i define as a norm to the plane of rotation of an axis. A rotating axis does not have to rotate in a plane but my definition of orientation as a fractal uses this special case to exemplify it. Simi8lar dwfinitions have been culturally used to define angle measure etc.,and without further elaboration i use a measurement fractal to measure axial extension, an angular measurement fractal to describe orientation rotation in a quantised form and also axial rotation or spinning. I further establish a foundational orientation and a foundational rotation called the cycle which is measured in radians as 2π by definition. see polynomial rotations
Of these three motion attributes of space orientation is the psychological one in that i as an observer choose outside of any system of axioms i develop a foundational orientation which i call and experience as "fixed" even if it is not or it is only apparent relative to me.

Apparent extension can be modeled in setFS by the relation
                         
                   apparent extension=k*ln(local extension)

which is to say that the static observer sees a different measurement to the observer moving in the direction of the extension at the pace of movement of the extension. In other words, the observer in the moving reference frame sees a different measurement to the static observer within whose frame of reference the other observer is moving.

The iteration producing the vorticular field of motion in setFS is not identified here but my intention is to define it in sub planck limit  space and or super universal space. However the motion fractally entrained in this iteration is thus by definition inertial.

The motion is also held in equilibrium by the fractal entrainment acting on a region in a coherent way. Thus the disturbance of this equilibrium leads naturally to forced behaviour and inertial or momentum transfer. This occurs not only in extension but also in rotation.

Just a quick note for further development. Taking the planck limit as the size of a "pixel", i will describe geometrical space as being a pixellated structure or organisation or arrangement. I propose that this is all that there is and so that there is no empty space in which this structure sits: just to be clear and logically finite. An alternative proposal would be to have this structure nested within an infinite fractal structure in a self similar way with boundarisation at iteration level boundaries. I would prefer the infinite iteration level model, but this logically implies that space is this infinite structure within which my geometrical space sits,but it also implies that it is not empty.

Whichever proposal i adopt, motion in geometrical space is through the pixels switching on and off or varying the energy levels they emit or absorb. This is radiative emission or absorption by neighbouring pixels. A collection of pixels cohered in some way will take on the organisation, properties and attributes of quantum space at some iteration level and then at some other iteration level the properties of classical relativistic space. The pixels at the planck level thus do not "move" in the classical sense but impart  movement to objects at a higher iteration level through boundarisation changes.

The "pixels" represent the measurable level at which the iteration process is "impressable" which is to say that my definition of any universal iteration processes cannot be distinguished below this length. This iteration process then i am thinking of as  occuring at sub planck lengths and the the planck length pixellated structure of the universe sequences this iteration process fundamentally, through a radiative process.

...but that interpretation removes Gaussian (or is it Netwonian?) invariance. By viewing the Planck length as a pixel size, you make certain directions behave differently than others. Really, the Planck length works more like the smallest distance you can measure... but then, you can turn 10 degrees and go back to where you came, and "probably" you'll be back where you started.

the planck lenghts seems to form a grid, but a totally isotropic one.
There is no such thing, yet known, in 3D though. - Well, yet known to me^^

However, I played a bit with spheres and based on the fact that an infinite dimensional sphere neither has an infiniteD Hypervolume, nor a corresponding surface (the general sphere volume and surface formulae have peaks at 5 or 6 dimensions or so and then steadily decrease...), together with an assumption for an 11-or-more-D Universe/Multiverse and the not yet found solution for higher-than-3D- close sphere packings, I had the idea, that maybe at higher dimensions, there is such a packing :)

Hello timeroot and kram1032 and anyone else who would like to comment or contribute. Your thoughts are very welcome. Please do not be afraid of the title maths as i pdsted recently, maths is a certain type of thinking based on a subset of language which focuses on iteration and the product of iteration namely fractals. The fact that i write this definition is due to the work of artistic people here and through the generations and throughout the worlds languages.

Language itself is a model of my experiential continuum and a model of my conscious and unconscious experiences. My thought today is about identity, that which i call "i". Simply put and to be elaborated at a later post : identity and interface are equivalent notions. Thus using the computer programming paradigms it is possible to trace the development of identity over time in the model of the development of programming interfaces. We recognise this 'identity' immediately: as a mac user i know a mac interface from a windows interface! Of course the interface itself and below/before the interface are collections of seething sequences of iterative processes.

What pops up on my screen is very much a symbiotic relationship between these iterative processes and the grid structure of the display. More relevantly the display algorithms and mechanisms that drive the display grid.

My point about planck length is thus illustrated by this paradigm: below the display grid pixel size and in a sense before and beyond it much is going on that i am not aware of directly, and the interface only gives me a coherent selection of the maelstrom that lies at the heart of a processor and memory configuration. I do not mean to rely on this paradigm,but to utilise it to convey a notion that is guiding my exploration. That is why i welcome any and all comments clarifications constructive criticsms contributions and collaborations!

The set FS is a model of notFS which i am building and exploring in an axiomatic approach, so i am looking for axioms to further the 11 or so that i have outlined so far. These represent my best guess at axioms and are open to modification.

For example i recently realised that i have not fully explored the basis for the SI units for time. Time for me is any periodic motion. By a fractal splitting method i develop a scale for fractions of that motion and by fractal aggregation i extend the scale.Quite naturally i transpose this scale into a geometric linear form,but often without acknowledging that this is a construction which may or may not have geometrical validity. The fact that it is culturally endorsed does not make it suitable for all uses. For example i am given to understand that the western concept of "time" is linear, while the indian and far eastern concept is cyclical or circular. With this conception i am able to trace the western notion of time back to persian Zoroastrianism through so called gnostic groups.

My point is everything is not always as it seems and thus it is valid to explore and question and propose and construct. In any case its fun too. ;D

Bootnote

Wen-Hann Wang may be the director of research for both circuits and systems at Intel Labs, but he made it clear where his sympathy lies when he said: "A circuit is at the bottom of the totem pole. It does all the work and no one appreciates it. The system gets all the glory."

I make no claim to originality, but rather seek to collaborate and explore. In this light the following reference highlights similar thinking about the foundations of Einsteinian space time (which term by the way i have traced so far to Wiliam Hamilton as originator). Because of its overt "mathematical" origin, i justify its inclusion in the setFS exploration.
  http://theresonanceproject.org/research.html (http://theresonanceproject.org/research.html)

My idea of planck length pixels forming an implied grid is just one possible outcome for sub planck length processes. It may in fact be a more kinetic structure for which a thermodynamic model would be more appropriate. However it is to be noted the fractal self-similarity in these suggestions, which is to say that a scheme for a larger scale is being applied to a different scale. How can i do other? and this again begs the question of the construction of all this fractality!
Never mind, i live in acceptance of my place in the scheme of things!

(http://www.fractalforums.com/gallery/1/511_28_03_10_1_20_02.jpeg)

 :groupbutts: That's what i am talking about,Willis!

This is my preferred fractal universe structure seen in a plane. In the universal void this would extend in any orientation one chooses and look like a great vortex. From the apparent centre of the vortex an anti twisting higher rotational vortex would jet out, and in the centre of that a higher rotational vortex might jet in or remain stationary. Each level of vortex will establish a coherent set of cylinders that will allow for stable universes of long endurance.

What was that bit about a range of frequencies to bring out the details?

I really wonder what paddelbrot could do with the 3D variants. He seriously mastered the Mset and several alternative formulae.
That one's just breathtaking!

Yes! Very creative images in his gallery!

The thought occurred that iteration requires an iterator. So to make that explicit for setFS i realise that the only natural/Yh¶h iterators i know are circular or elliptical or vortex motion. All of these are conical section curves so i especially choose the conical helix as the motion iterator for setFS as i think the other two periodic motions can be derived from it. i will propose this as an axiom for set FS When i have tidied it up a little.

The image of the fractally structured univese is so suggestive as it indicates regions flowing into other regions which are bounded iteratively by vortices. This to me is a powerful realisation of the big bang and inflation happenning within a local universe separated by turbulent boundary conditions from neighbouring universes.

A property of all conics is that they are surface-hugging. By this i mean that they lie in a surface that is conical, so i would expect a kind of coherency in their "action". I would expect them to form discrete layers for example and any spiral to gravitate toward a nearby surface to maintain ts behaviour, I would also expect them to be space filling at all scales while maintaining extreme coherency. Thus a system of circles will densely pack but still be discrete. This will also i suspect be true for a spiral leading me to suspect  that it is possible to have a spiral counterspiral arrangement which would enable the energy concentrated at the apex to be dissipated back down through a spiral in a densely packed spiral system. So perhaps the power in a maelstrom may exhibit this phenomena  .

What happens when these tightly packed and coherent conic section curves are intersected? I would expect the disruption to generate "conicsectional chaos" by which i mean seemingly chaotic curves which are in fact some polynomial combination of the conic secrion curves aith an additional impetus from the intersecting curve. It may be that a taylor or maclaurin expansion of the conic sections might reflect this or a fourier interpolation based on conic section curves may be possible to describe these chaotic curves.

So for example the boundary between one spiral and another will be a turbulent region of conic curves where the surfaces interact and give more than one possible surface to cling to. The variation in the density field at such a surface should be sufficient to entrain conicsectional curved behaviour proportional to ,in a dynamic mechanical system, the energy transfer between the the two regions in contact.

The combination of these conical section curves would look lie a vortex in geometrical space.

Just to note that given a conicsectional iterator certain things are straightforward corollaries. But first in my system there is no movement in space. Instead there is movement of space. Thus the conicsectional iterator promotes conic section motions. Thus these surface hugging motions are hugging movements of space that are conicsectional in nature. When i jump and return to the earth it is along a conicsectional curve. Einstein introduced the notion of curved spacetime, but i use a notion of a conicsectional iterator. All motion will in this view no matter how chaotic be some combination of conical sectional motion. Variation in density is involved and proportional to the amount of energy in the space under equilibrium.

If i applly this conception aximatically, that is to say as a first principle when analysing a physical event i get atural results in thought experiments.
So throwing a stone into water means the entry velocity and pathe will be parabolic, the movement of the water will instantaneously be conicsectional and should from this initial condition lead to conic sectional results in the first instance. I would expect the water surface to be pushed apart and the underlying water to be radiated downwards in conic sectional motions.

 The sufface would be impelled outward in an elliptical shape depending on how the parabola of initial motion lies on the cone. The circle-ellipse will be the intersection of the cone of the parabola with the surface plane of the water and will move in such a way as to realise this cones surface.

Consequently when energy density equilibrium is achieved by instantaneous viscosity friction forces the motion of the object will become either instantaneously zero or uniformmotion in the direction of greatest instability. The cone of the opened surface will invite conicsectional motion as it returns to equilibrium above the sinking object. Thus a pattern of spiralling vortices will radiate inwards and outwards at all levels .A radiation of concentric circles will form around the site of entry, high impact cylinders of material will form aeound the main instantaneous vortex and die away parabolically radiating hyperbolically. At the initial instance of impact as the water surface is first broken a hyperbolic compression wave radiates out depending on the angle of incidence, becoming ellipsoidal or circular the steeper the angle of incidence.

The conicsectional motions also generate horizontal vortices within the "splash wall" which dissipates the viscosity energy hypebolically into the surrounding denser material. The high energy instantaneous vortices on impact create a jet which shoot some of the less viscous surface up out of the impact site back along the parabola of entry

just now had  great thought that given a straightening out of mass/density and friction/viscosity and definitely removing the "nothing" absrraction for space, conicsectional motions and curves could combine elecromagnetism and gravity through variation in density and viscsity. Thus light for example would be high energy hyperbolic radiations in a rarefied density medium with very low viscosity. The density will be assumed at plank length sided volumes and the viscosity will be proportional to the planck length distances between between these densities.

Of course i have to take into account the redefinition of "time" that conicsectional movements imply when thinking about velocity etc on a conic surface. Conic sectional movements have a repetetive movement called a period. If i draw a "grenwhich" line from the apex of the cone tthen i can charaterise this period as a distance moved on the surface of the cone that intersects the grenwich starting from the grenwhich. This imediately distinguishes circles ellipses and spirals. Of the three spirals are distinguished by a variation in this distance due to the loop not closing and so crossing the grenwhich at a different distamce s along the grenwhich as measured from the apex.

Now to construct our current notion of time i need motion. In set FS i always have motion as an entrained result of the iterator the universal vortex. The final thing i do to construct time is to take a standard period at a standard value of s along the grenwhich and use the observed motion there to define a unit of time. I then use fractal processes to define my subdivisions and multiplications of scale.

What happens if i use a spiral as a standard period? The time i construct from that will have curious properties but what? and remeber i can not talk about standard speeds as ths is meant to define a standard, but i can compare and contrast relativistically between a standard defined say on a circle with circular motion.

These proportional comparisons would come under the earlier discussion about equivalences, and are a feature of the definition of standards. So mass for example as defined is in reality a standard density and force is in fact a standard pressure, but that is not so easy to see until i define pressure using newtons axiom for motion under force which has the word "impressed " included in it.

Now helical curve/motion has a special relationship to conicsectional curve/motion in that a helix intersects a conic surface in one or 2 places, and if we extend the conic surface through the apex then always in 2 places.  However there is a special case when a helix is tangental to a surface i the  cylindrical direction of the helix. in this case the helix can touch a conic surface in one limit point or along a conic sectional curve of limit points.

the other classical object that interacts with a conic section is a sphere and a spiral on the surface of a sphere. this again intersects a conic urface in a maximum of 2 places with tangential limit points of a circle or at least 1 limit point. I use the term limit point to emphasise the iterative nature of these observations, but all is in fact iterative .

Now the intersection of these surfaces with the conic surface in general is giving a conic curve or a pair of conic curves which are linked by the intersecting surface. These then are candidates for linked or entangled motions and probabilties and also tunnelling effects of electron  holes.

In my observation this also explains the observation of cylindrically stable regions inside a vortex. As a geometrical space object the surface of a cone rotating will entrain all internal cone surfaces. which i will observe as stable cylindrical regional motion which is a result of stable helical motion on a cylinder at that regions conic surface reference s along the greenwhich line. This will also be affected by the viscosity of the space being rotated, so that theoretically infinite helical motions will devolve into a chaotic cylindrical surface of a given thickness and same viscosity .         

The observation of tangential helical motion/curve to a conic surface also allows me to see a helical motion radiating out from a conic section impact event along with all the  conic sectional events. However due to the stability of the helical motion on a conic suface this is likely to be an extremely short radiation close to the origin of impact. The helical cylinders however are likely to propogate along the vortex wave propogation effect that radiates out from the impact region,decaying under a conic sectional curvr law, probably hyperbolic . 

I was going to put this inthe above post but i will isolate it to this one.

As mentioned i have thought that abstract concepts of force and  and mass are not accurate concepts fr the set FS, because i cannot define mass without volume, or force without surface area of application. For me Density and pressure are more applicable conceptons. a;ong with this i generalise friction to viscosity and generally adopt the fluid mechanical descriptions.

The thing about using this paradigm is that motion of dense objects in an equilibrium pressure field is towards regions of lower pressure, hus pulling a dense object in a pressure field which is in equilibrium is creating a small region of lower pressure into which the dense object is pushed. Depending on the viscosity interaction between the dense object and the pressure fieldis how much and how fast the movement occurs. The natural inertia of the dense object is then a combination of the underlying iterator entrainment and the equilibrium this sets up within the variable densities region and the viscosity of the pressure field medium the dense object is in.

Now if in general movement is toward lower density how is condensation possible? My only suggested solution is in the behaviour  of conic sectional motion. Providing a vortex motion exists at a centre in the lower density and that vortex motion is of a higher speed than the environmental vortex entrained motions the space will condense around that lower density core until an equilibrium is reached between the vortex energy and the surrounding low pressure field energy.

If this is the case then the core of a condensed object should have a lower density than the surrounding "medium" which in turn should have a higher density than its surrounding medium enabling an equilibrium/ stability wave to be established. Thus peak density should occur between the centre and the limit surface of the condensed object and should be susceptible to the conic section curve motions in the system.

Why in lowest density space do objects condense into a spherical shape given the the underlying conicsectional curve motions? My suggestion is that the shape is not  sphere but a torus but with a high speed vortex at the core making for a closure at the vortex tops/bottoms in a dual cone i apex arrangement with motion being directional in that condensing motion occurs on one cone system and evaoration/sublimation on the opposite cone system   balanced by an anti spiral interaction going the otherway dealing with the low density material in the system which arises from interactions of different densities "releasing" higher density material back into the void, but as a superfine spray or mist of enormous velocity.

These incredible densities with such enormous velociies are the basis in set FS of electromagnetically discrete behaviours, and are way way way below sub planck length in dimension which is one reason why we generalie them as field effect phenomena. I might add hat a candidate for aether they maybe,but aether is not a needed concept im set FS as i do not have a "nothing" to fill. All of these effects are just variations in density motion viscosity and conicsectional and helical rules of motion under a universal vortex iterator,pressure being a consequence of motion variation within and on the surface of a region. The basic axioms of geometrical space being extension along conic sectional curves, rotation around axes that point along or tangential to conic sectional curves with no fundamental orientation all being relativistic, and finally iterator entrained motion along conic sectional curves due to a universal vorticular iterator which generate helical and toroidal curves as tangent to or intersections with conic surface curves/motions.

According to Bayley, the target of this scene is projective geometry, a subject that involved concepts that Dodgson would have found ridiculous, particularly the "principle of continuity." Jean-Victor Poncelet, the French mathematician who set out the principle, described it as follows: "Let a figure be conceived to undergo a certain continuous variation, and let some general property concerning it be granted as true, so long as the variation is confined within certain limits; then the same property will belong to all the successive states of the figure."

When Poncelet talked of "figures", he meant geometric figures, of course, but Dodgson playfully subjects Poncelet's description to strict logical analysis and takes it to its most extreme conclusion. He turns a baby into a pig through the principle of continuity. Importantly, the baby retains most of its original features, as any object going through a continuous transformation must. His limbs are still held out like a starfish, and he has a queer shape, turned-up nose and small eyes. Alice only realizes he has changed when his sneezes turn to grunts.

As usual Taking the time to question fundamentals produces interesting results. This is the periodic motion our standard of time is constructed from. Of interest is the helmholtz coils used to trap the caesium particles: another spiral!

http://www.nrc-cnrc.gc.ca/eng/projects/inms/fountain-clock.html (http://www.nrc-cnrc.gc.ca/eng/projects/inms/fountain-clock.html).

So the iterator in the system is a stable quartz crystal resonator of 5mhz, so how are these mhz measured without a definition of time? Not obvious but the solutions discussed here are iterative solutions! Start with the old definition and iterate it to the new more accurate definition.

A feedbackloop is an essential part of the design and formally represents part of the iterative process within the design. the motion of the particles are conic sectional motions and the force used is clearly referenced as a pressure. The density within the system is controlled by a cooling system using lazers to effect the necessary condensing forces within the magnetic  helmholtz field which is spiral in nature. 

So watching Richard Hammond's invisible world with interest and noticed the raindrops and milkdrops sequence. Surface tension and viscosity are related. The spherical shape maintaining its shape at freefall speed depending on droplet size and air density and viscosity is interesting,as was watching it disintegrate when equilibrium conditions were exceeded. The instantaneous shockwave curve seemed to initially be a bezier type curve of a high order before being overtaken by a radiating conic curve going from a bezier suface to a conic surface. By this i simply mean that the underlying curves for thr observed shapes moved froma bezier description toa conic sectional one.
I noted that the milk dropped vertically followed a curve that was conically either straight on the conic surface or direct from the apex of a conic surface through  evry connectected entrained conic apex within a given conic surface. therefore iwas eager to see if cylindrical and helical motion ensued on contact with the viscous surface. Further analysis is required but it looked consistent with expectations, with the elastic bounce effect being entirely consistent with anti vortex behaviour at the instantaneous centre point of impact.
 The pressure shock wave was also evident in the water surface and quickly seemed to damp whether that was hypebolically or harmoniously requires further analysis.
I understand references to mass as a volume reference to a dense material and force as a pressure per unit area but wonder if pressure per unit volume might not be more in keeping? Although force is a derived unit as is pressure as is acceleration, and really is a quantification of motion and arises out of observations of motion, i feel that the conditions under which the observations took place are not fully reflected dimensionally in the definition of force if the definition of mass is changed as i advocate to a density per unit volume of some substance based either on Avagadros constant or a Planck unit.

Things in the set FS are ordered by the conicsectional ,helical and spherical curve behaviors entrained by the universal vortex . Anyone of these curves can destroy the order of a system if they move in a non equilibrium direction. In that sense order and with it life and computational consciousness is very fragile,and it is very reassuring to see a strong region of coherence in vortex behaviour.

i proposed somewhere on a physics site that the proper paradigmatic medium for light waves is light, just as ocean waves are in te ocean etc. So now the shock wave it seems to me represents a boundary condition for any advancing medium with idiosyncratic wave behaviour being either side of the boundary. The boundary in set FS is always a fractal structure and thus represents intensified energy conditions in this structure. I would expect light to produce a shockwave in its advancing edge and thus as is currently stated this advancing edge would travel at or above the speed of light with the following wave behaviour averaging out at the speed of light. The advancing shockwave is what would initialise all the properties of light such as reflection, refraction etc.

Of course this would be generalised for all electromagnetic radiation.

Now a question: are the magnetic field lines around a solenoid a helical torus structure? This is to say would a test electron hypothetically take a helical path around the solenoid passing through the open or vacuum core of it repeatedly as it traverses the toroidal surface ? And would the toroidal surface the electron travels on helically through the core, depend on the energy of the electron?

 

I want to redefine force as pressure per unit volume P with the actual definition being

           P=

Where p is the "gas" pressure and the is the measuring volume of the gas and A is the surface area of the measuring volume.

this then leads to
 
P =  (1)

For a constant pressure this has a solution

P= == =

Where is the "gas" density and is the observed mean acceleration of expansion in the gas which integrated over the surface area of the measuring volume will be a constant that is non zero but small. This represents an inertial /equilibrium solution with non equilibrium or non constant solutions resulting in resolved component acceleration.

by Boyle's law we can link  to the mean gas temperature, T:=k.T

I think that the simplest sequential method of integrating over a surface is by means of a surface spiral. Any space filling curve could be used.
The is an operator in this instance as it is undefined for calculation. As soon as a product can be passed into it it can receive a defined calculation procedure, in this case sum spirally around the surface area. As the product admits no change on an infinitesimal basis if is constant the result should be . However the mean acceleration is not a constant and so a small variation in the sum should be expected as the infinitesimal areas go to limit. However i would need to have a function for acceleration of expanding gas based on surface area expansion.

I want to redefine mass as density per unit volume D which is written in this way:

D= = ==

Where is Avogadros constant is the volume of the substance containing Avogadros number of elemental or compound particles as a molecular weight in grammes and is the measuring volume for the standard and molecular weight includes atomic weight for elements.
This is an iterative redefinition of mass so that the initial definition of mass for calculating the atomic weight is the current one.

The cultural iteration +1 is an example of specialism, in that it defines  counting as addition. A similar structure defines multiplication as counting in fixed value groups; +2,+5,.... being the 2 times and 5 times table numerals. General addition then is characterised by adding variable value groups and s called aggregation.

Similarly the iteration -1 defines counting in reverse as subtraction and -2, -5, .... as division which is subtracting fixed value groups and noting the remainder and how many subtractions took place. `in multiplication we note the final value and how many additons took place. General subtraction involves subtracting variable value groups and is called disaggregation.  

Now mathematicians have modeled a form of aggregation which is variable but in a specialist way : the variable groups are formed by  "multiplying" each variable group by the "group size", so these are termed power law additions and condense the notion of addition into a new process called multiplication by a base.

Distinguishing multiplication in this way tends to obscure the essential iterative base  of it so that it seems to be a binary operation when in fact it is a unary operation on a binary base operation of counting.

 

Binary operators require 2 elements to define there operation. A unary operator requires only "one" to act as a modiier. So the operator 1*, 7*,... makes no sense without the base binary operation +  Which instructs me or another to combine 2 elements by adding group size to a previous named element and to continue to do so until told to stop, calling out the names as i go.

Aggregation is specified by the 2 element names and a counting iteration starting at one numeral and continuing +1 the second element numeral of times. In the base operation the 2 elements define an iteration instruction or procedure one being the starting numeral the other being the iteration control numeral. This says stop after this numeral has been called out in the counting iteration while you have been continuing the initial counting iteration from the name of the first numeral. Thus 2 iteration counts are established the second controlling the first. This can be reduced to one counting iteration mod( the size of the aggregation of the 2 elements) . So 2+7 is not only 9 but also +1 mod(9) which is count or iterate forever mod 9 .

Multiplication operates on this binary base not on another element of a set, so in this strict sense multiplication is a unary operation on a binary addition iteration, groupsize . The elements of the set being operated on are usually 0 and .

Multiplication by a base is discrete as far as counting iterations go, as each numeral in the value name space has to be arrived at by its own independent counting iteration. There is no counting iteration that gives the sequence of numerals  in a power law addition sequence. Because of this power law sequences are useful as linearly independent basis vector forms or polynomial forms. This type of sequence of aggregation  products is a fractal iteration arrangement allowing independent counting to take place in each part of the sequence such that the power numerals record the cycles of the itereations in the preceding power numerals, ie 2 based power laws  can record  normal counting as cycles of 2 iterations.

The reverse of this leads us to look at disaggregation to find this special form of power law subtraction and in fact we find the basis for understanding this as a new process called division or better inversion. For inversion to make sense i have to have a constructed set of counting names that deal with the enumeration of splitting or breaking a boundarised object into pieces either by a natural destructive action or as a biological division process. This started to be attempted under the name of fractions but continues under the name rational numbers. The construction of this type of numeral space is a power law tour de force.

It seems clear to me,but then i have thought about it for a while, that Newtons laws are of motion but that motion is de facto  Euclidian which is why it is not as accurate as Einsteins which appears to be Riemannian.  The more general Riemann geometry is not hard as it is the geometry of plants and trees,and bulbous fruits etc. However we now have fractal geometry which is more general still and may lead to a more accurate description of motion.

As obvious as it may be i make he point that a 3d graphing programmw such as runiters focusses on the linking aspect of mapping while a 3d fractal generator focuses on the motion part of mapping. So the "surfaces" produced tell me a different story for each programme . Iteration is about motion and motion surfaces or orbit traps i think the correct term might be, and the fractal generators sculpt this motion. The simple rules for sculpting  highlight the complexity of the sculpted motion. More complex sculpting rules may be possible to describe growth motions.

If i think of the fractal generator as a video camera then it produces images of motion stopped at a certain iteration. The surfaces and edges are frozen in various states of motion. I have hitherto thought of these products as fractal sculptures,but am starting to see them as still images of a dynamic process sequence. At one time z^2+c was a polynomial in z a so called "complex number" but i term it a polynomial numeral. Now it is a predicate of an iteration statement z=z^2+c. And this statement i now see as a description of how quad numerals are to move under the iteration! and the quad numerals are polynomial numerals in 4 variables with a basis of 4 linearly independent power-like operators which behave like power nomials in a polynomial. By using unary operator analysis i see the relationships between the constructed basis and hope to further devise tools to explore this further, but it feeds back into the construction of mathematical operators in general, in particular the fundamental ones of addition, subtraction and division and multiplication , and base-logarithmic multiplication/division on which we algebraically construct arbitrary numbering /counting systems. These numbering systems are turning out to be part of the language we need to describe classes of iterations in the observed experiential  continuum. So splitting in cell division can be described, but the fracturing of a plate or a rock crystal- well not quite yet, but almost.

What drives this innovation and development of mathematical thinking? I think and believe it is iteration. The development of infinitesimal math at a time when infinitesimal thinking in science was derided as absurd by certain religious logicians(those that study persuasive arguments, and forms of confounding a proposition!) is a case in point. Derivatives and differentiation is constructed on the basis of iteration. The structure of these iterations are not difficult but are confounding if one is not used to iterations. I have heard that indian mathematicians relished infinite fractions, and through this understood one fractal: the limit boundary. The fractal limit boundary became crucial for Newton and Leibniz to progress with their mathematical description of motion by iterative mathematical products such as the limit boundary/value.

Of course summation and limits have always been associated with infinitesimal addition and this has developed into the integral side of calculus,using iterations to arrive at a sum. It is a wonder that so many of these operators are linked by an inverse relationship that can be demonstrated but not entirely surprising.

So the humblr fractal generator is in my opinion a fundamental mathematical tool for looking at processes of growth and destruction and condensation and radiation and a whole lot of other stuff.

(http://books.google.com/books?id=Tm0FAAAAQAAJ&pg=PA20&img=1&zoom=3&hl=en&sig=ACfU3U0DhX4KkHf9myeT-HbteaIHFNIY4Q&ci=257%2C467%2C677%2C907&edge=0)

Looking upon newtons third law i am impressed that the notion of "opposite" is not so firmly expressed, nor explained as being linearly opposite or "right" as Newton earlier expressed the motion arising from the impression of a force.Newton i have observed to be every bit as tentative and qualifying as any careful observer should.

Thus i have already redefined the laws in terms of vorticular motion. But it strikes me afresh to observe that the yin and yang symbol describing and designating two fundamentally opposing "forces" in chinese cosmology both small "scale" and large does not indicate a right action but rather a spiral one. This is of interest because the western notion of right action has been influenced much by the cultural transference of information and it would lead me to think that newton in proposing these laws already had infinitesimals in mind. Thus his notion of right was as an approximation to the "true" motion which newton attempted to divine from studies of circular motion.

 Newton was well aware of relative motion and relativity but not able to advance his studies to Einsteins level publically having no cultural need to or relevant data to inform a direction.

The chinese however for some reason not clear to me yet had observed this relationship between opposing principles at all levels and divined a spiral or vorticular relationship. I do know that chinese philosophers did not give credence to absolutes in the western philosophical sense and this may be why they shunned any absolute opposition. In any case the evidence for spirals and vortices in opposing interactions is manifold.

just a note for further development. Mathematics is part of language as a category and as an epistemology is a function of the general faculty of thinking which in its raw state is information processing.This requires the notion of information which inturn requires referents informer and informee. The informer transfers the information to the informee who receives the information. Now the referent for information can be deined as the particular state that the informer holds, inheres, maintains and demonstrates etc . The informee is the recipient of this state in that its state is modified by the receipt of this state. The informee necessarily has a state and is thus an informer also and may in receiving information transmit its information to the informer.These state changes may be as simple as confirmation changes in molecular structures for example.

With this definition of information a more precise explanation of transmission and receipt can be arrived at iteratively.

The utilising of this information as in information processing is several stages down the line of this structural concept, but represents an epiphenomenal product of these initial constructs. At this later stage parsing and syntax will become core concepts and fundamental to any iterative definition of perception. I also by default acknowledge the importance of the turing machine concept which will be central to the development of the structure of information processing. By way of these notions i hope to arrive at general language concepts and in particular mathematical language concepts as a fundamental language model of our perceptions of iteration in the set FS. By extension or discovery i hope to find more structure to notFS

A technical point , but of interest to me is the notion of Archimedian  groups and fields. These are fields that specifically exclude infinitesimals and infinity. Since i was clasically trained my notion of these conceptions have always been based on the notions of bounds and limits. In addition i have induced that the concepts of number and numeral are related but distinct and that mathematics or thinking about iterative processes and procedures and operators and algorithms to describe or construct the same do not require the notion of number, but rather the process of counting or numbering . The cultural iteration +1 being a numeral naming scheme is not infinite but is not bounded above, nor are rational numerals bounded below in the naming of any increasingly smaller fraction.

There is a clear disconnect between notions at this point but that is not unusual or indeed worrying in mathematical thinking that is to say iterative exploration. Discontinuity and boundary differences are normal in "fractals" and self similarity is crucial to identifying these discontinuities and boundaries.

In any case Archimedes was interested in quantization which is the basis of SI units, and for this you cannot have "infinitesimal numerals"  as such nor "infinite numerals". I acknowledge only 1 infinite numeral for computation purposes and that is infinity with
the recently declared numeral nullity as its axiomatic inverse.

Infinitesimals have always meant to me the limit process applied to a region in geometrical space as finer and finer scales are applied. In principle this is Archimedian because i practically have to stop when :sufficiently small to produce no appreciable difference in the result. The use of the term infnitesimal here means i have freedom to continue until such a situation arises. This can be well expressed in archimedian axioms and so leads to no difficulties at that level.

We could define all archimedian Quantity groups as clock arithmetics with various non integer  modulo so for example would represent the radian principle set of real values.

The question is can i live without and ? In the numeral namespace they clearly have a role so my guess is no. The flexibility it gives algebraically for notation is also not a mean advantage.

The notion of proof by contradiction also deserves a note. This really should be named what it is, that is: a test of consistency with the preferred axioms. Thus every mathematical model built with this test should avoid major additional inconsistencies beyond those introduced in the axioms.

Further it is worth pointing out that the models i build of the set notFS are always going to disconnect from and have a region of applicability to the set notFS, this in itself is iterative and leads to the development of fractal models of notFS.

 :rotfl:

I am laughing because in 1973 when i was at uni learning about math and computers nobody thought to tell me that the definitions of the number set R was changing to exclude infinity! In fact a lot of time was spent on Cantor infinities, which i rejected at the time or rather the then extant diagonal "proof". I rather liked Cantors infinities the countable and uncountable ones. The kind of "numbers" we used back then were called real but now they are called hyper-real ˆR. Now it is made explicit that a transfer rule applies from R to ˆR so i can do the things we used to do back then.

These technical distinctions mask the boundary between different models of the iterative processes in notFS and add the additional smokescreen of logic. This is not to say that i do not appreciate logic but rather that it adds nothing in credence to any mathematical statement, but rather tends to abstraction as Bertrand Russel and A N Whitehead  clearly demeonstrated.

Usually mathematicians will start with N the natural numbers assuming that these are trivial and well understood, whereas in fact these are not trivial because they do not exist. What does exist are the process of counting and the naming of each stage/ iteration of the count. What this cultural iteration may be used for is then up for grabs. We often play with it, console ourselves with part of it go to sleep doing it backwards and oh yes  we also Quantize with it! We can Order and rank with it and we can derive an algebra and an arithmetic for it . We can and do often derive mystical significance from the names we give to the stages, and we can and do attribute to it properties we derive later in higher mathematics! A bit circular i know, but what can i say: That is how it is !

So looking at not FS i see Aggregation, disaggregation, translation and rotation and perhaps the most important thing iterative motion. From these things i can construct a model that underlies all my so called mathematics, that is my thinking about all these fundamental iterative processes. I need a language to express all this in and that is why mathematics is a fundamental subset of all languages or means of communication. It therefore is not a subset of logic as Russel Fauningly thought, nor is it a Philosophy or a meta physics. It should be a playful engagement of our senses in contact with notFS and our languaged response to that experience, And how does that differ  to a writer or a poet or an observer of nature? My contention is that it does not except in the style and density of the notation and the utter repetetiveness of the subject matter.

I wonder what names we shall agree to call the infinitesimals when we realise that since 1973 we have had these "numbers" at our command. I bagsy "oneeta" for the least upperbound for the infinitesimals ;D

I have learned something today courtesy of phractal phoam phil: solidity does not inhere in space it is attributed to space.

I explain this anecdotally. When i was learning about science iwas given examples of solid liquid and gas to help  form the concepts of solidity liquidity and gaseousness. This of course seemed pretty straightforward and comprehensible. Then i was taught atomic theory. the result of this was that i reviewed my understanding of the liquid and gaseous phase in the following way : solidity inheres in atoms; the liquid and gaseous phase are explained by the kinetic theory of atoms. Thus solidity remained unchanged and in fact spread into the liquid and gaseous phase by attribution of spacial properties via the kinetic theory. Then i was introduced to high energy physics and Niels Bohr's  work and solidity was gradually pared back to protons and neutrons and electrons and massless photons, with the particle wave distinction gradually blurring the description of matter. Solidity was replaced by probabilistic description of position and the newer particles of quarks and eventually colour Quantum Dynamics(QCD). By this stage i had simply forgotten about the concept of solidity having reached out for a concept called "energy" which being indistinct and not well founded remained the linking concept for all new results from high energy physics research. I might as well have called it "Bleh!" for all its use, but it sounded /seemed scientific and utilitarian. However it was not falsifiable and so is in fact tantamount to a faith statement. Nothing wrong with faith statements but they do not tend to advance scientific enquiry.     The fact that it was an ill formed conception i now realise because i could not understand how "space" could be solid. Even though i have briefly flirted with a pixel description of space unrelated to the planck length i could not clarify how that related to motion in space etc, other than by a switching on and off of pixels. The energy conception of space would have plasmas as exemplars but again solidity is inhered in some particle or other.

What dawns on me now is that solidity does not inhere in a particle it is an attribute of space in that it relates regions of space to one another through relativistic properties and not inherent properties of a particle. Thus space can be in constant motion but the relative motions encode relativistic properties which are the attributes we appreciate as rigidity liquidity and gaseousness. This does not require a substance other than space itself but any substance should exhibit the same properties by fractal entrainment to the fundamental properties of space in motion relativistically.

In the set FS i have set an iterator as a fundamental universal  motion which fractally entrains all motions in setFS. This fundamental iterator is a vortex. That means space is spinning universally in some fractal pattern in setFS. The spinning patterns consist in vortices in infinitely various arrangements size and boundarised regions. It is these arrangements sizes and boundary conditions along with fractal entrainment in which all the the attributes of solidity, liquidity and gaseousness inhere along with plasma phase. Energy then becomes synonymous with rate of spin of the vortex and vortices , that is energy is proprtional to ∂²Ω/∂p  where Ω is the angle of rotation in radians and p is the period length of 2π radians in metres.

The fractal entrainment and coherence within the vortices is the "medium" through which conic sectional curve motion is transmitted through the vorticular space and arises as tangential motion to conic helical motion within the vortices. Also arising out of the vorticular motion and interaction are loxodromic spiral motions giving rise to spherical and toroidal curve motion within the sururface of a conical helix vortex system.

This is entirely testable within any medium capable of sustaining a vorticular fractal pattern.

Note on computational consciousness. The computational refers to the manipulation of the molecular and cellular structures within an organism or a structured environment . These structures are at least fractal in that there are structures within structures at different levels of investigation and of different level of dcomplexity.

the arrangement at the dna /rna level allows for a manipulation of the molecular structures through electrical, chemical and hydraulic pressure, and gas pressure feedback feed forward interactions.

At the cellular level the arrangements and connections of cells provides a cellular substrate for a circuit board analogue which provides for an electronic cybernetic system to be expressed, while similarly hydraulic and mechanical cybernetic systems are also established through cellular structure and manipulation. Inherent within these structures is also a chemical messaging system with microbiological inputs and outputs which also form a cybernetic system. These feedback feedforward systems interact so that the hydraulic and gas pressures within the structures and cell structures alter electronic properties within the systems which are then fed back to cascade other alterations in a cybernetic way through a mechanical biological structure in tandem with chemical cybernetic systems .

Without complication the nervous system delivers the main identifiable structures for the computational systems but this is in tandem with the other structures within the  organism. The senses of visual, auditory,gustatory(smell and taste) and kinesthetic/proprioception  are global descriptors of this systems inputs and output systems, which necessarily are complex and iterative and fractal in structure.

Although computational consciousness is a neat phrase it is no different to information processing consciousness or data processing consciousness .

The iterative nature of computational consciousness is clear and the products of that iteration are fractal perception and and construction and concept development. In addition fractal structure is evident in all products of the organism as a whole in particular reproduction.

In order to rethink Geometry i am coining a phrase Spaciometry.

Spaciometry will be the study of the forms, boundaries and surfaces, and structures of and within "objects and structures in space" from the point of view of computing properties of equivalence in and between those objects and structures. It will be foundational to spaciometry that iteration and iterative techniques will be clearly identified and utilised. It is hoped that this may reveal a referent for "fractal geometry" which encapsulates the idea of "roughness" as Benoit Mandelbrot originally had a predilection for. Meanwhile i will start with Riemann's Hypotheses and go from there.

i have been studying the spaciometry of my garden and it is interesting to look at and derive notions from. Topography is naturally apart of spaciometry and the topography of the structures and objects in the garden is a treat in itself! The topogahical arrangement of forms and structures by juxtaposition  or sequential patterning or relative patterning and structural arrangement is fascinating . And the arrangement of structures within structures and the asymmetry and asimilarity is refreshing. The relationships of growth in structures to the similarity within structures is interesting and the notions of boundaries as applied to complex structures made up of well defined objects which themselves are structured is very fractal. The notions of surface continuity discontinuity regions and spatially oriented objects which are distinct but connected is also exciting . The notions of space filling and dense yet not solid  are also fascinating.

These forms appear by a growth process which is a combination of internal dynamics within the structures modifies by external conditions, very much like the operations within a fractal generator. I can also see the dynamic disaggregation or erosion or degradation of less dynamic forms by dynamic boundary conditions.

It is interesting to note that Euclidian forms simply do not exist because there are no Euclidian lines of points. Even man made objects exhibit a cobbling together of Euclidian ideas which in one explanation of Riemannian geometry makes these object representative of Riemanns hypotheses.

Measurement by equivalence is strongly suggested and twigs and nets of stalks seem natural measuring tools. Orientation is flexible as some structures are rigid and others are motile, and some follow the sun. Notions of twist and rotation and translation are evident and tessellation of a surface is hinted at but indicative of the difference between living and non living  or slowly growing forms. Convexity and concavity are exemplified and many polygonal forms are in evidence as well as polyhedral. Truncated conical forms f varying heights are in evidence as are loxodromic folding and other spiral/vorticular forms for plant stems and the like.

There are no planes only surfaces and consequently 2d forms are an abstraction. However there are very thin 3d forms which have a consistent depth over a surface region which will approximate a plain. Cartesian and polar coordinate systems seem a natural mesh like measurement tool with variable orientation mimicking the natural rotations in some structures. I can establish from each of these types of structure an extension measure and a rotation measure. I can also establish a density measure as a measure of iteration or tessellation  which is commensurate with notions of surface area. I think that a notion of space filling is easy to establish but volume is less obvious because density is so intrinsic to space filling but volume attempts to exclude density. The common notion of volume it is apparent is a surface area bounded space, which is a generalisation of area which is the edge or boundary enclosed space.

Just found that my notion of conic sectional curve motion is on the track that others have been exploring before me. So i am pleased that i have adopted it for the set FS. It along with my latest axiom on (energy) motion allows me to think of gravity as the motions defined by these curves.

In spaciometry i have a chance to define a spaciometric density, which is an apparent notion and axiomatic to spaciometry, but i have to go back to axiom 1 to define it.

The construction of my experiential continuum utilises inate processes and procedural forms, hence my interest in axiomatising parsing and syntax, which i will get round to. It also uses inate forms in fact form is an inate notion derived from as it turns out all the sensory inputs. Form is a synesthesia. We have an extensive-model of the world which gives location information not only by binocular vision but by bin orificial sensation! So location is given by the visual kinesthetic auditory and gustatory and proproceptive sensual systems through processing algorithms in the brain. Kinestthetic systems properly include proprioceptive but i am distinguishing the external from the internal. Interestingly our kinesthetic includes th whisker effect of the hairs on our skin, which also contribute powerfully to location information .

Thus my primary reference system of orientation extension  and rotation relative to the orientation is a conglomeration of the sensory input processed by an evolutionary and possibly revolutionary algorithm process in the brain which the whole organism accepts s a fundamental stabilisation . However alaong with these stimulus response products are also products such as mas, density, surface, form structure etc. The notioins of region and boundary are such products where region uses orientation and a soft focus to give a meaning to its denotation. The soft focus provides a boundary impression without defining a boundary. Boundaries can then be defined or discovered in this region in terns of sharp focus and the particular rods and cones in the eye stimulated by this procedure and aor any other binary type output from the supporting senses.

Mass then refers indiscriminately to the substance being sensed that is space itself. I can now quantify mass by boundarisation and form and structure and surface. Mass can be defined by a numeral representing the count of the structures and forms  and surfaces and boundaries identified within a region

On the face of it a smooth surface may count at 1 form and/or 1 structure within a defined region, but i have to be careful not to confuse abstractions such as planes etc with regions. A surface is only usually one face of a form and special forms should be looked at in their actual not ideal format. Currently i can link mass through Avagadros constant to molecular structures in a form. Spaciometry includes these atomic forms in its conception, but this really highlights the fractal nature of all forms and regions and the structure of a region when boundarised may be analysed in this way to determine a mass.

Smooth surfaces are interesting  because they highlight another aspect of mass and that is extension. This is why there is always some confusion between volume as it is defined and mass. the notion of extension is utilised as soon as i boundarise a region ie put a boundary corresponding to cone and rod signals from the eye that are binary. Thus i will modify the notion of mass as i explore it more.

I also have to look at the density of surfaces and structures within a boundary to appreciate and determine mass and kinesthetic contributions have to be included as well. Mass as a conception is fundamental to my notion of space and abstracting from that a mechanical definition has i believe divorced me intuitively from space itself!

I have yet to consider gaseous phase matter in terms of appreciation of its mass. Although not strictly gaseous i will begin by looking at clouds.

The profound result for me in developing a spaciometric mass and density is the discovery of magica trigger that gives me a full synesthesia of mass combining the notions to reaify mass.

Considering the gaseous phase of matter and in particular water vapour and clouds i realise that these regions of space have colour, from invisible in the near locality to opacity in tone and rainbow hue. This variation in itself gives me information about mass and density of the gas in the region. If i then include the contribution from smell i begin to realise how pervasive, subtle and powerful the apprehension of mass and density is. simply by smell i can appreciat the mass of a gas in the region i am identifying.

The spaciometric concepts of mass and density have many contributory sensate notions that will need to be combined in any definition or denotation.

The notion of space has to be realised as a matter or substance, a material that is elemental and exhibiting of phase states and regional attributes that are relativistic motion attributes. Regions of space will have a spaciometric mass and density.

The mass i will denote by the extents of a region's boundary surfaces, the count of internal boundaries, structures,  internal regions, surfaces, forms,and the variation of the auditory signal, colour and opacity and strength of smell within . The region's defining boundary surfaces are not assumed to be continuous or contiguous and in fact may be an imposed or iterative approximation, by axioms 1 to 5 of the set FS. In addition surfaces are equally definable by focal lengths and magnification parameters as well as extension. All surfaces and forms boundaries and structures are computational products of processes within the central nervous system and are fundamentally iterative. Thus spaciometric mass is an iterative product of a computational CNS process.

Spaciometric density is the numerous structural internal relativity of the forms, structures, surfaces and boundaries within the defining boundary surfaces of a region. The numerosity of the relative internal structures is enhanced by the colour and smell variation and the auditory soundscape. In density it is the structural relativity that distinguishes it as a property of mass. Thus for a gaseous medium the relativity is clearly over a greater extent than for a liquid or solid phase. Also the motion of a gaseous form is relativistic and is more apparent and diverse than that of the liquid and solid phases. The structural relativity means that density is always apprehended relative to the region it is appreciated in. However i can recognise congruent or similar regions by colour or smell or texture (variation of colour and structuraland auditory form and kinesthetics) in which the same structural relativity is apprehendable and this will lead on to constructing through equivalences a fractal scale for density and then mass.

What i have found for me is that spaciometric mass and density is synesthesiastically reified by the kinesthetic and proprioceptive senses. Thus i apprehend the spaciometric mass or density of a region "magically" by contacting it! Thiness, gaseousness, fluidity of mass is apprehendable by this and indeed comprehendable! It also gives me insight into why "weight" or balance equivalences were considered originally in the definition of mass for the SI units.

While the smell aspect of mass may seem a little odd it is a real contributor to the notion as does the auditory signal, which assists in orientation and mass boundary determination through a sonar sense.

On another point it dawns on me that in spaciometry i ought not to assume that pythagorean forms will be the standard. Rather these will be special forms. Thus my "definition" of conic curves and motions will have to be generalised accordingly. Now i read that einsteinian relativity uses geodesic to classify spacetime curves so i will use spaciodesic.

I have to mention one thing in passing, cos i am currently thinking about the "angle" in spaciometry as an imposed cultural norm or abstraction that though an extremely useful tool, has its drawbacks and non sequiturs.

the form a=b^2 has its shape determined by the reference system or rather the reference frame devised for the parameters. Suppose i use an orthogonal set of axes      C(x,y) with x and y being the parameters of the points in this frame; compare the shape with a parameter frame P(ø,r) which has no orthogonal axes but uses a rotation parameter and an extension parameter or B(l₁,l₂,ø), which again has no orthogonal axes , but rather 2 line egments that cross at an angle ø with the parameters measured along tese line segments.

the same form traces out distinctly different shapes:

y=x^2 being a parabola curve, x= y^2 being a rotated parobola:

ø=r^2 being a spiral curve, r=ø^2 being a spiral curve rotating and expanding differently to the first.

l₁=l₂^2 being a quadratic Bezier curve shape varied by ø.

If the reference frame can fundamentally determine the shape i discover i feel that formulae are not of themselves a shape. They are a relationship between the parameters which we can explore by reference frame by algorithmic procedure sequence and most relevantly by iteration.

The iteration setup for a=b^2 is b=0:a=b^2, b=b+1.

There are various implementations in code for this.

The plot function is where the parameter frames can be implemented , the algorithm for b^2 is hidden but can be b*b, b+b+...... b times, b mod(2π)*2 , a taylor maclaurin expansion etc.

The iterator can also be b=b^n+c, b= bmod(n)+cmod(m),z=z^2+c etc including any series expansion form,and any calculus form such as x=d(x)^n/dx+c, or x=dⁿx/ds+c  etc where the parameters have to be in there calculation form not the manipulation proforma which itself is iterative in expansion to the calculation form.

By manipulation profrma i mean the symbolic notational form which is then used as a procedural guide to the calculation form. I will elaborate on this distinction in mathematical notation and symbology at another time, but suffice it to say these static forms lauded for there elegance obscure the dynamic often iterative process of manipulation and oerator action required to arrive at the calculation form which is itself an algorithm enjoining action by the reader or processor. these levels of required action are what make algebra and math so daunting, because they appear to have no referent.

Spaciometric mass and density i have outlined as counting procedure to apprehend its "quantity" aspect or appreciation. However, as basic as counting is it is a foretiori to measuring. Measurement is fundamental to sensor arrays of huge numeral dimensions, and measurement is essential comparison of equivalences. Whenever i want to count i set up a region as a standard and measure that region against others which i then count off as they match. The match criteria can be anything from exact congruency to a specific item.

It is this measurement that is the basis for counting and all quantification and this is the connection i will pursue to the comprehension of mechanical mass.

I have been working under the apprehension that my primary language response to the set not FS would be naming and numbering; from which i would at a later stage proceed to show how mathematical thinking chiefly in its algebraic form would naturally derive with counting as a specific application of an algebra called in full arithmetic based on notions of natural numbering which is to say the culthural counting iteration +1. This counting iteration is in fact a specialised form of naming, that is numeralisation- the naming of quantity by numeral names, such as "one", "two" etc..

It is the notion of quantity that goes unrecognised in this explanation of the derivation of mathematical thinking. The notion of quantity is a notion of measurement. It is the notion of measurement that is fundamental to the language response to not FS, so it is naming and measuring that seem to be my first response and the measurement is an inate response. 

Measurement then i consider as an essential apprehension and comparison of quantitative signal output of a sensory "array", where the "array" is itself a very large quantity of individual sensors each one capable of giving various signal outputs. At this level of detail it does not matter if the output is binary trinary or indeed analogue, because the individual response is subsumed within the overall structural response of the "array" and regions within the "array" .

Thus the structure behaves in a manner as follows:

The individual signal has no meaning, as it is in no context to give it a meaning. Two sensors providing 2 signals are the basis of a comparison, but require a combining sensor to record the comparison. So the immediate sensing level passes signals to a comparison level, and the fundamental structure of this arrangement is 2 sensing to 1 comparing.

Although i can immediately divine a binary structure forming, it is a structural binary system not a sensor that outputs a binary signal.

It iss again clear that the structure has to be iterated up another level to give meaning to the comparator level signal, so a fundamental structure would be 4 to 2 to 1. At this level of comparing the comparators the system is able to hold individual signal information, and comparative information. If i iterate the structure one more level then the system can hold information about regional response at the 2 sensor level and can compare at the 4 sensor level as well a individual sensor information: the structure is 8 to 4 to 2 to 1 .

This structure of sensors can hold information as individual signal as comparative signal out put as regional signal output and as comparative regional signal output, as well as some other combinations, for example comparative comparative signal output. However, without some detailed diagrams this would soon become confusing and i am merely noting a structure as an example.(good news, see below).

It is clear that the structure is an information system, but what is not so clear is that it is a self referential information system, and that it acts by measurement /comparison of raw signal, that is by equivalences. The system clearly has emergent properties at each iteration and as you might guess self reference is not the top emergent property. Self reference occurs at a lower level than the structure goes up to. By this i mean that for  the 8 to 4 to 2 to 1 structure self reference occurs when the system compares the 2 regions below a given level. So in the 2 to 1 comparator level there is potential self reference but it requires the 4 to 2 to 1 structure to have self reference at the 2 to 1 level. This means that the structure holds information about its state but cannot describe its state true state to "itself". However it can precisely describe its constituent states up to the level where the self reference/ comparison is occurring.

Now suppose the structure had 2 comparators, by which i mean the 2 sensors feed their signals via 2 separate structures, say a second comparator that connects directly to the top level of the structure. This would give the whole structure a direct comparison feed which could allow the structure to have an "internal and "external" representation of the same information, thus allowing the system a freedom to be "creative". This means the system could measure the information in the structure an assign for example a " truth" value to it. These kinds of sensory structures are relevant to the description of innate sensate notion forming processes, and thus to the topic of consciousness and of course information processing at the thinking level.

If the description above is valid then it is also valid that my initial response to not FS will be measurement. Thus measurement would come before language, and at the level of language that humans are at would become a forgotten or mis-described foundation to our conscious response to not FS. It would lead me to say that "non verbal communication" is the basis of all language and that this NVC is primarily measurement of sensor responses. Thus my appreciation of my experiential continuum is based on measurement and apprehension of comparative differences in those measurements. This would then lead "naturally" to a naming response to those comparative differences and a numbering response in terms of quantitative differences. The numbering response would be in terms of "instance". "instances", the "one, two, many" response reported in many so called primitive numbering systems. I am using numbering here as a precursor to "counting", and as the process of "naming numerals". I have distinguished between numerals and numbers for reasons of rigour to avoid as much as possible the mystical attachments to numbers. This is not to say that i am against mysticism, but rather to be clear what i am referring to. Thus numbering as a response precedes counting which is the cultural iteration +1 and which is performed by rehearsing the numerals, that is the numeral names given to us through a cultural "numbering" process, which more directly put is just a cultural numeral naming.

It is clear to me that measurement processes of an iterative nature in an iterative structure for sensors have given rise to the impuls to name and number, and from this we have culturally derived a counting iteration which has been the basis of our algebraic apprehension of quantity, which we formulated into various arithmetics. But at the same time we continued to respond to our measurement impulse to extend our appreciation of quantity in terms of  orientation, extension, boundarisation, relation, motion, form and structure, a well as other comparative differences, such as colour, smell, kinesthesia and audition.

(http://www.fractalforums.com/gallery/3/410_26_08_10_3_36_15_1.png)

I offer this image from the Internet Encyclopedia of Science (http://www.daviddarling.info/encyclopedia/R/retina.html) as an illustration of the sructure discussed above. One ought not to think that this structure is unique to the eye. The sensors are unique to the eye in the body i believe but of course not biologically unique as this is a general vertebrate eye form.

As a restatement of some initial discussions earlier in the thread i revisit the notions of quantity and quality and express these notions as products of a measuring and distinction process or set of processes that occur from the sensors to the central nervous system within the organism that inheres "my" experiential continuum.

The sensor "arrays" do have a spaciometric input which is again an information source about the spaciometry of the experiential continuum. Vision contributes to the notions of form, surface, orientation, hue; audition to extension, orientation. pressure, sonar, balnce and rotation; gustatory contributes to regional identification, memory for place, kinesthesia contribute to locality awareness, extension, motion , movement, balance, motion transfer and a grounding "presence" which synesthesiastically combines and comprehends the whole sensory map. This is just some of the result of the measurement and distinction processes that inhere in the relativistic forms that structurally combine to form the organism in which my experiential coninuum inheres.

When researching the foundation of calculus i find a catalogue of misconceptions and missed opportunities.
 The situation is not unusual however a s mathematicians a re children of their cultures and time. There are several precurdors to the notion of calculus but the topic is formalised by Leibniz and Newton in 1670's on the basis of a dispute. The nature of Leibniz, a poly math and a high achiever versus Newtons more secretive and mystical nature lead to an unsteady start to the topic with weak foundational axioms and posits. Berkleys  challenge to Newton's calculus sparked off an intense revision of the foundations that supported or supplanted the notions of the founders Leibniz and Newton. In the course of this new notation experimental at the time was tried out and developed and accepted on partisan grounds rather than sound communication principles thus d²y/dx² though familiar is a confusion of signals.

Euler's patient revision of the symbology , methodology, and pedagogy did much to provide an acceptable explanatory schema of the algebra of the developing calculus and enabled mathematicians to revise the foundations in a more coherent way .

I hope to explore the precursors and the founders notions to see why such a powerful working process delivered and delivers the goods despite seeming to be ill founded.

The Greek notion of proof or rather the classical notion of proof is the first culprit, and the religious sensibilities of the time are the second- for if you sincerelu hold that there is a realm of man's enquiry and the rest is god"s perogative then venturing on the infinite and the infinitesimal is a great impertinence.

All things greek though admired were not wholeheartedly accepted, so challenging the god of the roman catholic church was not a culturally encouraged thing, and many of the priest intelligentsia made it their duty to point out errors in all things scientific. For this reason and due to his autism and sexual ambiguity Newton was very private about his ideas. Heknew that roof under the greek system was by trial. Trial by our peers who may have personal axes to grind. Leibniz knew also that logical arguments, as a lawyer are designed to convince, not t prove in the modern sense of some mathematicians. Logic in fact is the study of argumentation to convince or demonstrate the inadequacy of a notion, person or thing. Chiefly to discredit a person was a valid and accepted form of argumentation and proof, on the basis that the gods would vindicate the truth.

It is therefore of paramount importance to apply the dictum of Karl popper that Falsifiability is necessary and sufficient to defend the utility of a notion. Proof by contradiction or the "impossible" proof in fact only supports consistency, it doesnot "prove" a thing. The only sense for the word proof in mathematics that makes any useful contribution is trial to falsify..

Both Newton and Leibniz used a notion of vanishingly small; this was a value so small that when it was squared it became 0. These values came to be called infinitesimals. Now it makes no sense to us who are heirs of the "reai number" system after Dedkind, that squaring a small value would com to  exactly 0, but for Newton and Leibniz the Rational Numbers and logarithms were all they had, and it was possible to see that a very small number vanished when squared. It got so small that according to Newton it behaved as if ir were 0, and logarithms of these numbers produced values that were exactly 0 to 4 or 6 decimal places in a log calculation.

Newton went on to apply this notion to rates of change of velocities as they moved on some polynomial curve, which he called fluxions, but he did not explain what they were as he had no idea which is why he gave it a made up name. Leibniz at least was able to account for his infinitesimal elements.

The notions that guided Newtons thinking was the compound interest process, and its relation to the binomial theorem, the  notions of polynomials, the notions of extrapolation and interpolation.

It strikes me at the moment that Newon's laws of motion are validated by him by an intense study of motion, and Leibniz had a deep interest in and study of motion. Both hit upon the use of the binomial coefficients to describe or framework their exploration, based on the then complete model of growth and rates of change afforded by the study of compound interest. Nobody studied the nature or algebra of time so intensely until William Hamilton in his Theory of couples 200 or so years later,
 Therefore i am not suprised that modern theorists like Einstein and his wife have combined elements of the two in their descriiption of motion. In fact Hamilton's quaternions have proven to be a succint and powerful way to describe space time motion along wih newtons analysis of motion.

The combinations of the calculus and hypercomplex algebras has proven very useful especially since Euler"s Formula.

My study of calculus foundations is proving fruitful.

So i note from Wikipedia on Ratio"
History and etymology   Look up ratio in Wiktionary, the free dictionary.


It would be impossible to trace the origin of the concept of ratio since the ideas from which it developed would have been familiar to preliterate cultures. For example the idea of one village being twice as large as another or a distance being half that of another are so basic that they would have been understood in prehistoric society.[6] However, it is possible to trace the origin of the word ratio to the Ancient Greek λόγος (logos) appearing in Book V of Euclid's Elements. Early translators rendered this into Latin as ratio, meaning "reason". However a more modern interpretation of Euclid's meaning is more akin to computation or reckoning.[7] Medieval writers used the word proportio ("proportion") to indicate ratio and proportionalitas ("proportionality") for the equality of ratios.[8]

Euclid collected the results appearing the Elements from earlier sources. The Pythagoreans developed a theory of ratio and proportion as applied to numbers.[9] The Pythagoreans' conception of number included only what would today be called rational numbers, casting doubt on the validity of the theory in geometry where, as the Pythagoreans also discovered, incommensurable ratios (corresponding to irrational numbers) exist. The discovery of a theory of ratios that does not assume commensurability is probably due to Eudoxus. The exposition of the theory of proportions that appears in Book VII of The Elements reflects the earlier theory of ratios of commensurables.[10]

The existence of multiple theories seems unnecessarily complex to modern sensibility since ratios are, to a large extent, identified with quotients. This is a comparatively recent development however, as can be seem from the fact that modern geometry textbooks still use distinct terminology and notation for ratios and quotients. The reasons for this are twofold. First, there was the previously mentioned reluctance to accept irrational numbers as true numbers. Second, the lack of a widely used symbolism to replace the already established terminology of ratios delayed the full acceptance of fractions as alternative until the 16th century.[11]
[edit]
Euclid's definitions......

And from the dictionary

com·men·su·ra·ble  (k-mnsr--bl, -shr-)
adj.
1. Measurable by a common standard.
2. Commensurate; proportionate.
3. Mathematics Exactly divisible by the same unit an integral number of times. Used of two quantities.
[Late Latin commnsrbilis : Latin com-, com- + mnsrbilis, measurable (from mnsrre, to measure; see commensurate).]

Their are many notions to draw from just these two quotes alone, but i will confine myself to pointing at the greek apprehension of the measuring response inherent in our neurology is conveyed by the single grapheme logos and the phoneme"logos". As far as i can tell this is a uniquely greek apprehension and thus combines the sensory stimulus of visual , audio, gustatory and kinesthetic sensors. Thus logos is a full response that does not differentiate measurement and distinction processes, "compare and contrast" processes,sensor comparison circuit processes. It is thus ground zero for the greek response to the set not FS, and from it the language (Logos= word) and the measuring(logos= reckoning) response is drawn and subsequently differentiated by application of the word logos to particular contexts. It is interesting to note that comparison the fundamental root activity of measuring is drawn out in the term ratio by earlier translators into latin, and then by later interpreters into the term reckoning, which though a different word is hardly a differnt notion. However it has more potency for thought for non mathematicians. And this of course is the sad thing: Our specialist use of language has divided us as did the mythological event at the tower of Babel. Nevertheless i am content to proceed on the basis that my measuring response is the basis of impulse to distinguish by quantity and name, that is to say by "logos".

I then hope to advance the fundamental nature of iteration in my whole experiential continuum in the development of the "logos" response, and thereby to facilitate a intuitive response in mathematical thinking as well as language study to the fractal products of these iterative processes.

One thing stands out already: the fundamental nature of ratio and its development: proportion, to an intuitive facility with mathematics from spaciometry to abstract symbolic manipulation. In fact the loss of ratio is very much akin to the loss of rationality in mathematics!

Keeping in proportion then, the fundamental nature of Equivalence classes is noted, and the practice of thinking in ratios and proportions is to be encouraged, by which i mean the method of positing or establishing a ratio and then by induction equating ratios to establish a rule or law from which a proportion can be derived for each instance. This encompasses not only exactness where it may be found but more importantly relativity wherever the ratio holds. Thus by ratio exploration it seems we can explore not FS by induction and deduction, and tus we can intuitively interpolate a proportion and even extrapolate a proportion once the inductive rule for a ratio has been established. Hidden within the term induction is of course the iterative nature of exploration of instances from which i may derive a rule and then by test of interpolation accuracy and then extrapolation accuracy determine whether it is a law for such parameters and contexts which i am currently investigating

The falsifiability of this approach is evident and thus satisfies Karl Poppers admonition on proofs.

I say these things after noting that earlier mathematicians had this facility to think Rationally and then to derive proportions as needed. Having had sight of Newton's papers for the first time this is what struck me as most significant.

I venture to add that Newton in his earlier explorations in 1665-1668 developed such a facility for rational thought through training , particularly by John Wallis who directed his studies, That by this method he was able to perceive and develop the generalised binomial series. And from intense consideration of the problems in algebraic geometry at the time he fused this binomial series together with tangent theory under the ubiquitous compound interest methods of Schiller to form geometric calculus. He then rethought this in terms of motion and curves of orbital motion, and happily found that the same rules and methods applied to motion. Because He derived the same rules from fluid motion elements he called these fluents, and the limit tangents fluxions. He was not working with algebraic geometrical curves, but with fluid(continuous) motions broken down by the geometrical method of decrements into small fluents, the limits of which were the tangent of the motion at that point, which by compounding(extrapolating by the binomial series expansion) using the compound interest methodology, enabled him to integrate the tangents into the curve traced out by the motion in its entirety.

This was an insightful moment based on his facility with rational thinking, enabling him to proportion the area under general curves and from considerations of tangents(ratios) to extrapolate(integrate) the curve. Of course he had to note some funny ratios called infinitesimals thereafter, but the binomial series inductively supported that these were tending to 0 and that some even were zero! That i think was more a reflection of the limit of tables and accuracy of his time.

I do want to look at polynomials in the light of our decimal system being an instance of a polynomial called a geometric series, because the development of operators and algorithms in transforms have a fundamental shaping effect on the polynomial concept and may have a visible accelerating effect when the decimal and arabic form of numeral formation were adopted widely. In any case it provides possibly a secure foundational theory base for a revision of number theory along iterative lines.

My knowledge of polynomials however will need to be refreshed by exploration.

The binomial series alone in my opinion validates Newton as the discoverer of Calculus with Leibniz in the role of a fellow collaborator rather than plagiarist. However Leibniz ambitions are suspect and he may have published early to beat Newton and for the advantage the recognition gave him in his own country. Leibniz you must remember seemed to survive by patronage, so his needs would affect his ethical considerations.

I woke this morning and thought about laser light diffraction gratings which are variable and tuning. This often happens and i feel it is an effect of the computational nature of my experiential continuum. This manipulations at a so called quantum level will play out the same response in all systems tuned to that sympathetic resonance frequency. Some will get a clearer signal than others,and some will be able to act on it while others will assist the manipulations enabling them to persist and extend. If you want to call them ghosts in the machine you can, but essentially they form a kind of software encoding of a continuing distributed universal computation .

The http://en.wikipedia.org/wiki/Virtual_retinal_display (http://en.wikipedia.org/wiki/Virtual_retinal_display) and the basic hardware http://en.wikipedia.org/wiki/Grating_light_valve (http://en.wikipedia.org/wiki/Grating_light_valve) pretty much nail the concept that i had come up with, happily. The notion itself is therefore not originated by me which i do not at this stage in my apprehension of my experiential continuum expect to be the case. Rather i expect there t be some coexisting creative development of an idea that i am perceiving or formulating. This is because as we communicate many seeds are scattered which grow as weeds to some and as beautiful flowers to others, but they grow in a multitude, rather than in isolation. I see this pattern which i have used in the several metaphors and paradigms i have used so far in this post, in the bacterial growth behaviour or in mitosistical cell division.

Because the idea is well established with regard to laser light diffraction gratings i can now advance  a "pixel" for the diffraction gratings. Physically this would look like a diffraction grating in which the diffraction slits are crossing each other, let us say initially orthogonally. The actual slits are what allow an interference pattern to emerge on the other side, showing the wave nature of light. In this case the light source(s) would be white laser light in the form of a beam. Now it occurs that if the beam is rastered then variations in the interference pattern will occur due to the solid angle from the point laser source, and this would have to be compensated for, but essentially a set of parallel slits should produce an interference pattern local to the slit. The emissive radiation would thus be modulated, and perceivable by the eye. I might then see the expected oil-film rainbow patterns. Now add the modulation of the laser beam as per a televisual signal, what would i see?
Now add to this the orthogonal diffraction grating arrangement or some similar arrangement of parallel diffraction gratings and what would i see?

I would expect the "square slit" to have a radially emerging interference pattern locally behind it. This interference hemisphere i would hope would behave like a "pixel", and would with a modulated laser beam raster produce distinguishable images in or local to the diffraction grating.


So now i am going to find out who has done this at any frequency level.This may then provide me with a robust model for quantum interactions at the planck distance, modeling the spaceometric structure as a diffraction grid.My ultimate goal however is to model a relativistc motion grid with diffraction grating properties.

Of course the 3d diffraction grid would have to be explored built up of elements of thin diffraction gratings. Already i can see that the signal attenuation would be marked as the diffracted light moves from layer to layer, but this is consistent with the darkness of the generally.

I also thought about ratio and proportion the day before in terms of the way my neural network of actual axonic nerve impulses records and manipulates my apprehension of the set not FS and the set FS, but i will detail that in another post. The central nervous sysytem i see as a distributed processing neural network with a central coordinating processor, running a base operating system at the nanozoe level with additional software installed by a mentoring process called parenting and socialisation.

Looking again at the foundation of tensor analysis and the work of Ricci-Cabastro and Levi-Civita i realised that Einstein like Newton stood on the shoulders of many giants, not to mention David Hilbert.

I have given a lot of thought to Ratio and Proportion in the development of the "logos" response and see the unifying nature of this impulse and in particular a qualitative difference in mathematical thinking before the modern period. I now suspect that many of the innovations in the development of maths derive from the ratio proportion type of response and thinking whereas mathematics today seems to be about remembering formulae and equations and how to manipulate them symbolically, developing axioms and so called proofs that lead to an internal consistency in mathematics but very few new conceptions because it has all become too damn abstract!

"Abstract" means looking at the essential transform, that is algorithm or operation,and working with that as if it were a real entity.

"And YH¶H spoke and it was so!". Our language system has a reification operation embedded within it. It can bring something into a kind of "existence" simply by naming it. The archetypical example is the word "quiz", but what the etymology of this word shows is a reification process for all languaged notions and concepts, which may not be based in a reciprocal apprehension of not FS, but rather a new reciprocal apprehension within the set FS .

The Tensor notion is unsatisfying in how it is communicated because it is couched in abstract language and not interms of everyday things.

However our world has changed by the introduction of cameras and LCD displays and VLSI circuit wafers etc which enable us to give a referent to a tensor which is concrete. In fact  "concrete block" is a tensor model which can be effectively utilised to ground the notion, or even better a "lego brick" block. A matress is another good model of a tensor.

Any way let you know if an other things "spring" to mind! :embarrass:

So i am on the way to the dentist and i think: extend the tensor notion to include spaciometric forms which are structured including "arbitrarily" structured forms eventually, but think of a plants cell structure in cross section, to begin with.."

Of course the spaciometric form is invariant under basis transform, that is to say reference frame, and spaciometric mass and spaciometric density can be attributed to the tensor.

Then i heard that because England scored some goals and won a game the economy was estimated to respond in various sectors "memorabilia, electrical items,sports items, FMG... to the tune of about a billion pounds. I saw a "complicated" input oitput structure there which could be represented in a tensor.

Then my dentist actuall y drilled my tooth and i saw a tensor relationship there between the input drill speed and path and the spaciometric result due to chemical and physical bonding energies being exceeded and strain maxima being exceeded being representaable in a tensor which would amalgamate the physical and chemical effect of the action of the dentists drill. The information stored in the tensor being accessed in various ways specific to the required output information.

Tensors as you may be seeing, although a spaciomtric form, may hold relational information, that is relativstic information, in this structured form. The structure of the tensor itself can mimic the structure of the form that it holds the information for or it can just be the "block" form with the correct structural relations adhered.

The fundamental relationship of a tensor is that the form does not alter as the reference frame (basis) alters. Thus if i reflected the components of a tensor in a "mirror" i would fundamentally alter the form of the tensor without realising it as the form looks the same just reflected. This would not be a tensor transformation as currently defined and needs a development in tensor algebra to accommodate it.

I am used to reflections as mathematical transforms but physically they have no counterpart except in the notion of chirality(left and right handedness). Whereas i can relate a left handed version to a right handed version by a mirror reflection i cannot physically do the transformation in the "solid" phase. These transformations will typically be found in chemical reaction products in the liquid,gas or plasma phases.

This relates to a discussion i began with regard to unary operators defined as spaciometric rotations of orientations and how a reference frame with an additional rotational attribute to the axis vector transforms a referenced region on its transformation.

In fact i proposed a more comprehensive vector definition, but now think that a tensor form would more than cover it! ah well such is life! ;D

iimagined that funny effect you can get in those halls of mirrors taking place physically: The left side of me would squeez into the middle and then "pop" out on the right side of me and vice versa, but it would be an anti me that results! Everything would work in the opposite way! Kind of weird science!!

Any way for that reason i say reflection is not a tensor transform even if it is by Ricci and Levi! Except as i noted in the area of reaction products. I suppose i ought to allow seeing what you look like in a mirror for those of us to whom it is a daily necessity. Go on then, put it in but don't say i did not warn you! :D

My current interest is in trying to define a ralativistic frame of reference, and when i do i will have to make it tensor invariant also.

A tensor is a quantitative model of a spaciometric form that is firstly relational and this enables it to be relativistic, secondly differential and this enables it to be mapped to a network of related reference frames and finally quantitative in each differential element of the model of the spaciometric form in a relational way.

Although this is a one line staged explanation it is still not a comprehensible denotation. It is however more apprehendable.

We utilise and view many possible candidates for a tensor in our modern computing environments: any spreadsheet aplication allows me to create a tensor as a spreadsheet for example, where the spreadsheet is the spaciometric form and each cells represents the differentials which are thusly relational  and referenceable relative to a common origin and also relative to each other. Thus a network of reference frames can be created that enables the whole form to be referenced by the origin/common  reference frame or by a relational reference frame. This allows for macro and micro relations to be "describable" and is why i thought that tensors might be a relativistic reference frame initially. They are not, but they possess a relativistic reference frame, which i have just described.

Finally each cell can contain quantitative data which is corresponding to its reference either relativistically or by common reference  or both. And additionally the quantitative data may be related by function/formula to any number of cells within the spreadsheet. It is to be noted that the quantitative data may be measurements or ratios of any quantifiable property of the spaciometric form at that referenced region

Now spreadsheets are a specific instance of a much more general class of applications which are: relational databases. Thus any relational database can be constructed to form a tensor which will have the description already advanced. Some relational database apps do not have all the mathematical functionality of a spreadsheet but this is a design consideration not a fundamental distinction, as this can be easily added to any relational database.

It is of course possible to reference between spreadsheets collected into a book format and across books et al .

Thus a good example of a tensor is a quantitative relational database. Many scientific measurements of experimental results are held in such relational databases, which when constructed appropriately would be exact tensor candidates .

So iguess just to finish the thought, that 3d Fractal generators are some kind of Tensor sculptor and tensor manipulator algorithms with the formula being the relational link to all differential elements of the tensor. These differential elements at least in Quasz are related in the form of a quad block.


So what i am delving into now is a spaciometric understanding of rotation which is not based on the euclidian tool "angle" but rather on some more topological notions.

I am fairly certain that i can distinguish two types of rotation which are discontinuous to each other but linked by topologically based limit processes, In addition i want to begin the analysis based on the "logos" response to see how much easier that makes intuitive mathematical thinking. All sensors are involved but the two i think i have to use in a TOTE analysis of spaciometric rotation are a visual region or boundary as a mrker for axiomatic visual orientation and an auditory linked sonar map with the auditory canal orientation sensors switched on. A Test Operate Test Exit (TOTE) is a basic , the basic cybernetic algorithm for traversing a self regulating system. This also happens to be the most basic iterative process that can be devised. Thus iteration and comparison and distinction are at the foundational level of any self regulating conscious intelligent system.

Again if this means my apprehension of the fractal nature of everything is inevitable because of this or that this is evidence that the fractal nature of the iterative processes in the universe are universal and always acting at every scale i cannot possibly distiguish.

  That is way intensor than I thought it would be. 

  I didn't read the whole thread (maybe at a further date), the 3rd page, than 62% of the 5th, and perhaps most of this one, so don't know if you mentioned it...  Kaluza Klein theory could be considered one of the precursors to modern multi-dimensional theories (not that it isn't modern, as it is post-relativistic, but perhaps pre-Copenhagen).  I ended up, long ago, coming up with it on my own (based on reading about relativity) and then mentioning my idea to a physicist I know (slightly) who told me my idea sounded like Kaluza Klein theory (which really isn't that hard to come up with, if one has a basic understanding of trig and special relativity).

  So I named this quick paper I threw together (a long time ago) KK math tricks... and I know you like math tricks.

kk math tricks at google docs (http://docs.google.com/fileview?id=0B0EcJQ49B_yOMzliNTJhNTYtZDE3Mi00NTJmLTkzOWYtNjUwOGI3ZTdjZGYz&hl=en)

Hiya matt, got to run but great to have some input. I am hoping Charles Wehner will return and give his insight. Will read your post link later.

Just to note that if i use P(ø,r) where r= n*ø and ø is in radians i have a reference frame which i can apply to a plane using 1 parameter ø and scalars n which are elements of the hyperreals as another parameter.

So now i am waiting to see if i can intuit a frame that works spaciometrically like this. This is inspired by plant leafing patterning, how plant spatially organise their leaves. Yep, i have been in the garden again! :D

So i was going to write about the probabalistic nature of my experiential continuum, but aside from the fact that it is a personal response to the uncertainty of induction and prediction it was only to note that our culture has measured it by chance, luck, likelihood, and probability and gaming odds. This combinatorial and permutational ratio map has brought the binomial theorem and the binomial series(discovered by Newton) into an important juxtaposition with a more computational approach to probability and a wider application of the probability model to physical/motion events, that is relativistic events; provided they can be represented by a compound interest type formulation. Thus  second order Taylor Expansion with all its conditions provides a basis for a probabilistic formulation of the event described by the initial formula .

any way i guess i should flat out just define a unch of terms related to the logos response. These tems are all based on the comparison of sensor signals as measured by a iteratively structured array system with massive parallel processing functionality, i.e. me!

So i am going to have i think visiometry, audiometry, gustatiometry,kinesiometry which has an important subdistinction proprioceptiometry. Some of these words are pretty ugly huh? I might come up with some cooler ones later.

So basically there is in each of the major sensory systems in the central nervous system/ central neural network a process of ratio distinction and storage, which constitutes the comparison or measurement response of that system to raw signal input. Thus an initial sensor array has its raw signals processed and stored as a ratio response: the signal at its sensor locations compared to all the sensor locations and their differing signals. Immediately the concepts of Area , Focus , and attention become nascent and emergent along with relativity in location. All these factors as well as the raw signal are stored or inherent in the structured array system, available for the system to utilise at different levels of processing and iteration.

This is the logos response technically desceibed an the above terms enable me to focus on describing the functioning of one of the systems or a synesthesia of the systems.

When i listen to music the fact that it is an experience of ratio variation is immediately apparent,but when i look at a visual scene the symphony of ratios in it are tuned out because that is a musical metaphor. a patchwork quilt of ratios comes to mind. And the sensations of texture and touch are a "thrill" of sensational ratios of hard, soft smooth; just as the gustatory responses produces a bouquet of flavours all ratioed to one another.

~It seems obvious to me that our logos response is  ratioed response that is the source of all our language distinctions and grammars and syntax as well as our mathematics.

A bit of fun which may help me to apprehend a relativistic reference frame.

How to build a binary tree. (http://mitpress.mit.edu/sicp/full-text/sicp/book/node34.html)

And then

How to build a hanging binary tree mobile  (http://www.wikihow.com/Create-a-Calder-Mobile).

Some homework (http://docs.google.com/viewer?a=v&q=cache:Cz2ExIV9HfEJ:www.cs.arizona.edu/classes/cs433/fall09/assignments/hw4.pdf+how+to+make+hanging+binary+tree+mobiles&hl=en&pid=bl&srcid=ADGEESjlfMcSrQjIYMmP2d1lhj44zXw7M1yr631_lCWURtx5uRb8L7RunvPG-c16dt9w4JU2H8W5pTISq61Rgkcn8tCNtD7_vS-mNDNi2iSYGXSWzAP-zYUtSOAZqQ9h4memy1WslgAU&sig=AHIEtbQgFohU69qkSYArHjUEUCgatpT81Q)

Which leads to a computer storage and processing analogy (http://en.wikipedia.org/wiki/Binary_heap).

Which almost gets to my idea of a binomial tree mobile with the notion of a binomial heap (http://en.wikipedia.org/wiki/Binomial_heap).

Here is an application of a binomial tree to stock price evaluation. Essentially it is a tensor for a stock market movement over a given time frame (http://docs.google.com/viewer?a=v&q=cache:r6VDeirBEZYJ:finmath.uchicago.edu/new/msfm/current/Programming/Binomial%2520Trees.pdf+how+to+make+hanging+binomial+tree+mobiles&hl=en&pid=bl&srcid=ADGEEShmOd0uFRTunPCBQkh2tJXjwIEDec8w7jSyn6gZo0Cy2Ninfss2cN9uFTuN3-WlRZOi6PRlYpJT1NY88z_hbIg6NYjwCk6GIiPxE1b7FxfycRj3WF3iOtMGzPAk3_hGfr5lhlHn&sig=AHIEtbSOuG7ZFjDSWN91lGhluO-HUJehOA). It is a good example of a dynamic tensor which can map dynamically scale variations in the quantities of interest, and show that a tensor is a store of relationally linked differential quantities in a relational way and does not have to have a spaciomeric form linked to it. In fact the tensor "form" as an array is a convention or set of conventions because the form for an array is physically different as one moves through different ways of representing it; eg. on paper,as a solid object, as a computer memory allocation, as a mobile... :tease:

So finally a binomial tree mobile can be simply made by hanging two binary tree mobiles from the same yoke.

My question is do i make a trinomial tree mobile by hanging 3 binary trees from the same yoke?

Have a play and enjoy ;D

When thinking about the history of polynomials, as such there really is no early history devoid of the general equation solving that seems to start early with the development and expansion of civilsation. But it is noticeable that such equation solving has a common history with matrix history and thus my feeling that the more general notion of tensor is a suitable  vehicle to frame the development of many related strands of mathematical thinking, arithmetic algorithm formation, and geometric notions in algebraic form.

The logos response enables me to see that the tensor has been at the back of all mathematical quantification of the set notFS; and the piecemeal development of the setFS has a unifying factor in the Tensor notion and in the logos response notion, which essentially denotes everything as a ratio distinction: My language response is essentially a collection of ratio distinctions with syntax rules and grammars that reflect the ratio relationships perceived through my CNS, in interaction with the set notFS. Because My CNS is an iterative processing system, the products of this system are fractals stored within the system in the set FS: a distributed memory system which of course is accessed iteratively and produces fractal results to consciousness (have you ever had that tip of the tongue experience? that is fractal consciousness).

Thus the CNS system holds many ratios distributed in a tensor form which themselves have been derived from structural array sensors in a tensor format. which are massively parallel processed to produce a tensor output fractally, which tensor is invariant under certain processing operations; Chiefly the affine transformation operations; but which same tensor of tensors is updatable and malleable to the iteration of notFS and set FS.

Thus when it is said that our initial response to the world was to name its elements and then to number them, i do not see it in that order. Rather we measure the world and name those distinct measurements which in fact are ratios. The development of mathematical thinking then has been the development of thinking language, and only when the language was develoed to specify quantitive information distinctly by numeralising it, did the syntax of language take a commensurate course to reflect the grammar of these numeral quantities. Our early history shows great mysticism around the concept of number because number was a name not a numeral. Over time through successive development and revision of specific language and then notation and the feedback feedforward loop between language and notation, specific mathematical lnguage and grammar and syntax was isolated. But because it is the logos response it was not isolated to die in its own obscurity! Rather in an intensely human response to notFS the language scope and facility and function of the mathematical thinking process language, was extended by exploration, and above all play.  

Diophantus (3rd century AD) is sometimes called "the father of algebra" in greek mathematics but it was With the collation and dissemination of the knowledge of maths at the time by al jabr"s author Muhammad ibn Mūsā al-Khwārizmī (c. 780-850) that modern maths arises. He learned Indian mathematics and introduced it to the Muslim world through his famous arithmetic text, Book on Addition and Subtraction after the Method of the Indians. He later wrote The Compendious Book on Calculation by Completion and Balancing, which established algebra as a mathematical discipline that is independent of geometry and arithmetic, with his parameterisation model of explanation, based on indian thinking at the time, in which even a colour could represent a parameter( hows about that you colour algorithmists!) .

Descartes was eventually able to begin the process of notating the tensor. With that it soon became possible to notate the difference between scalars and matrices after an extensive development of yes you guessed it polynomials!

Researchers now are able to identify the early matrix form in china and japan and the early binomial forms even the binomial theorem in india. Each instance of these forms is language based and purposed for a civilised, cultural outcome, so in a very real sense Descartes notation began the release of mathematical language and notion from purely group or cultural concerns with their mystical overtones.

However modern practitioners have gone too far the other way and so lost contact with the societies in which they live leading to the danger og mathematicians disappearing up their own backsides! (That is a topological joke if not a topical one) ;D

So really now we can trace the development of the tensor to LEVi and Ricci in its full blown modern form, and not before time too as Einstein needed Levi to tutor him in Tensors to be able to describe his Theories. Up until then one supposes his wife was more than adequate to the job!

So What of Newton and his Erstwhile collaborator Leibniz? My view is that had Leibniz not been so far away in terms of communication times that they would have been direct collaborators and even friends, but as it was Leibniz received his information second or third hand or late, but could  have known also from their source (http://www.angelfire.com/md/byme/mathsample.html),  the brilliant but secretive Newtonian mind. In fact it almost seems that had not Hooke mentioned a notion of his, to find the orbits of planets by compounding tangents, that we may not have ever had Pricipeae Mathematica!

Newton and Leibniz indisputably established polynomials as a secure mapping tool of the real world that is those elements in FS which have to do with change and motion in notFS. Thus polynomials are the basis for the success and accuracy of calculus when it wants exactness rather than ratios. But calculus is in fact all about ratios. The differential elements that support these ratios are in a one to one mapping with the differential elements within a tensor form

That to me leaves William Hamilton in this brief synopsis of mathematical development. As an engaging algebraist Hamilton in his theory of couples put the complex plane and all of complex mathematics on a sound an rigorous basis. He then went on to extend the basis to the third dimension and found he could not do it satisfactorily because it always seemed to demand a 4th basis element. Thus he solved it for 4 in a general way and got 3 as a bonus. Hamilton was seeking a description of time, and consequently and subsequently proved his theorems without the use of Descartes coordinate system, thus showing that the field he was in was unaffected by reference frame transformation, the defining characteristic of the modern tensor concept. Hamilton although the first to coin the phrase space time was to early to define a tensor as it is now thought of, but who really knows? Just a many topics :Finite differences, Power series expansions, infinitesimal numbers; were thought of and utilised by Newtn in his quest to describe the motions of the planets accurately, without elaboration or even publication, so Hamilton may have such notions unrecognized in his works because they were a means to an end not a subject in itself to him.

Relativistic reference frames. Just to note that these reference frames are exhibited in social and group rule behaviours found in the herd instinct behavours, group dynamic behaviours in gregariously organized group structures, and small group interaction models for space filling.

Specific examples are: filling seats in a bus, or positioning of individuals using an open toilet in which positions are clearly delineated by design (eg individual urinals) or not (troughs). Cockroaches in their gregarious social structures are good models for liquid and gas phase  dynamics, whereas the social structure of bees would model solid liquid gas phase dynamics. The general herd behaviours in fish shoals, locust swarms, bat swarms, wilderbeast and other animal migrations model liquid vapour dynamics.

Now the important thing is that some scientists and companies have developed algorithms to model this dynamic in the larger groups which have a high accuracy, and are impressively predictive.

This suggest to me that the Cartesian Tensor (cartesian coordinates), Polar Tensor and the Hamiltonian Tensor are reference frames that need slight tweaking to become relativistic reference frames, but even then may not be flexible enough to describe liquid and gas Phase. There are the more general Clifford Tensors to mention and there may also be, eventually a Musean Tensor if someone can work out what he was on about exactly.

 I am proposing that  a binary or binomial reference frame may be more applicable. However Binary or Binomial Tensors are unlikely to be affine transform invariant and so i will call them Flexors and Growors!

Flexors should help to model liquid, gas and plasma phase dynamics and Growors should model Growth dynamics of flexible non lattice structures, i think.

Just a note about other Tensor reference frames: Spherical, Cylindrical, Toroidal, Tetrahedral, Conic Sectional etc.

It looks like Dirac's Equation is going to be quite a journey!  :o

Have to say that reading other peoples musings highlight the difficulty of communicating with others. This is why playing together and collaborating is so important. The marks we scratch on bits of paper or tap into computer screens are just marks. The wonderful thing is that by sharing time together we can share a common reference for these marks. So I think it behooves all mathematicians to be kind playful souls full of patience and endless barrels of strong liquor! A kindly disposition did no one any harm and is a force for good.

It seems to me that Dirac and Hamilton have one thing in common an ability to explore abstract relations algebraically as let's and couples are essentially the same things.

Hamiltons quaternions are a positive development in mathematics because to me the method of their derivation is based on Hamiltons abstract algebra of time which he called couples. In this exploration it is clear to me that he tried to look at two related events and the algebra that governed those he derived by exploration. So all the while he was looking at relativity, relatedness, relationship. He used the line or vector to denote this relationship and some the inherent geometrical ideas which are thus independent of Cartesian constraints to place complex math on a secure footing. However I think that despite his wish to be free of Descarte's reference frame he used polynomials in a way that implies a reference frame just not a Cartesian one.

We cannot avoid a reference frame or geometry/spaciometry when thinking algebraically or mathematically. This is one of the unavoidable consequences of the logos response

I made a mistake. I explain it this way; I took a look at where i was going! Where possibly could all these ideas be coming from and leading too? :scared: I found Schroedinger, and then Dirac, and now Feynmann and his group, Add to that Haramein and it is inescapably looking like it is some kind of theory of everything? :hmh:

But this is in line with the Axioms, so why the mistake? :vomit: I don't want to do all their math! :embarrass:

Do not get me wrong, i am sure it would make an interesting read, and would even educate the pants off me :educated: but i don't want to do it that way, so i won't. I like going in my wifes garden and seeing the spaciometric ratios, the tensor reference frames and the topological phenomenon that catch my eye. For example Dirac's string "trick" [ realise is a topological function of spirals especially hose pipes. Why do they always kink when i roll them onto a garden hose spiral? I call it potential twist which becomes actual mechanical twist if the ends of the hosepipe are moved apart in opposite directions.
http://www.youtube.com/watch?v=CYBqIRM8GiY

When i found this video i was stunned to think that an idea i had was actually one fully worked out by others! So i know i am on the scent and i will follow it down to the core, but i will let others do the notation in their way cos i am too bogged down to learn their way as well as discover my way.

My thoughts on this and vortices are well laid out beforehand, but in addition i see how these systems can represent "pixel"s in a plankian lattice structure with soace deformable in this way representing on and off in the sense of a density energy tensor for this pixel. Or put it another way when the vortex is at its most definite and strongest the pixel is on as a dense form of matter, when at its weakest and least definite the pixel is off as a vacuum energy  form. Spin thus is functional as a determinant of matter phase state or as i want to call it relativistic form.

I am only writing this because it is apparent that the wave particle / wavicle debate is now a dead duck and we need a way of describing space so that it can exhibit all possibilities. The Quantum field theory purports to do this and my only observation is that the field is a substantive one  that is in vorticular motion continuously and in what seems like a quantized way, but really is a result of the binary nature of our sensor systems in making comparative distinctions. By this i mean that the cycle time for my sensory system to recognise and compare is such that i will only compute a sampling of what is happening in notFS and not an analogue. This is a generalisation of the uncertainty principle applied to compute cycles for information processing. As a result i am only aware of the set of information that my sensors have sampled and on the basis of this i have responded with my measurements and comparisons and distinctions.

It would seem apparent to me that many of the issues in high energy physics will have their natural counterparts in the digitisation of signals for inputting information into a processing system, and indeed the binary storage systems look attractive to me as more general reference frames for relativistic motion tensors .

Here is Feynman explaining what i am feeling!

 http://www.youtube.com/watch?v=lr8sVailoLw

I was going to write about the logos response in the context of an intuitive notion of tensors, but my mind has just been blown away by an alternative algebra and some topological representations of a "space" its properties and motions in the space holding information that changes. Then there is the starting notion of representing geometrical shapes and directions by a contiguous notation and developing an algebra from the geometry. It is called  higher dimensional algebra.
Any way it put me in mind of Hamilton a congenial and loyal algebraist of the 19th century and particularly his theory of couples  and conjugate functions. There is no doubt that Algebra was viewed in his time as a source of meditiation rule of thumb and formal language expression. The logos response was felt but not idenified, but Hamiltons 3 part distinction is revealing in that it describes the shape of a mathtematics athe time: Arithmetic, Analysis, and an Algebra of symbolic manipulation and notations which was a language exercise, just like grammar.

The logos response in my mind simplifies everything. There is a not fs there is an interface of sensors arranged in some system capble of responding to interactions with notFS, and both storing information about and imparting information to notFS. The stored information is stored and processed within a set FS which is defined by the iterative processes in the sensory system interacting with not FS.

The primary response of the system of senses with the processing within thew sytem and the memory store is the logos response: the system measures by comparison of responses from individual sensors in a binary or binomial or even polynomial very very large array stucture or set of structures;it then distiguishes by contrast(same or not same) and this gves rise to a "language" response. The language response is the notion of the interaction with the set notFS and is a pure ratio. What the ratio signifies is relative to each individual, thus it is relativistic form the get go. The vocal expressions i give to this ratio is what i call words. Some of the ratios are about spaciometric  considerations, others are about spaciomtric actions and still others are about details in spaciometric forms stuctures surfaces, boundaries etc.. certain uses of this language have been fashioned as tools to identify relations , objects, plurality and actions. Further specialism have lead gradually to the identification of the action of proportining , placing a portion of something pro to another portion. That is placing a quantity pro to another quantity. This is completely general : placing q pro to w. pro means before or next to or by. Q by w then is a proportioninig of quantities and it was called a ratio. A ratio of quantites. It is also a relation of quantities, a model or ideogram of the spaciometric relationship of quantities, an ordering and even a ranking, an array, an arrangement, a sequence.an ordinal with each quantity being a cardinal. And yes even a tensor.

Ratios were usually arranged into equivalences that is proportions were lined up or arrayed ingroups or classes each proportion(ratio) being in a ratio with the other proportions defined by the equivalence, the smallest quantities in proportion that have the same proportion as every other proportion gathered into the class.

Well this is neat. Lying in bed jotting down ideas.
Sa ratios were proportioned too especially if they were equivalent .this notion of equivalence is an inherent function ofbthe sensory system structure through comparison. Physicallybratios are different but a similar pattern goes into the proportioning of equivalent ratios, it is this similar pattern of activate which is used to define the equivalence.

So 4pro8 can be packaged by bundling 4 lots of 1 pro 2 together. This bundling is of course multiplication but it is built up from adding ratios so I have both addition and subtraction and multiplication of ratios right off the pat providing the ratios are the same, and repeated subtraction of the same ratios is the same as an operation called cancelling or factoring . Now factoring is a precursor to a more general operation called division. In fact providing the ratios are equivalent  can show that the equivalence class is an algebra group called a field.

So now we have a algebra with an obvious extension into non equivalent ratios.

A less obvious structure of the equivalence classes is the polynomial,but the great curiosity to find the rules and structures that enabled the "solving of proportions" would eventually lead to the discovery and reorganisation around these ratios which we call polynomial series and polynomials in general.

Notation lagged behind facility because language took time to develop these distinctions first orally then scripturally/pale ontologically. Only when the first civilizations arose do we see the development of scripts. Before that signs landmarks on various surfaces can be found which definitely show the oral languages were distinguishing ratios, which is not suprising as this is the logos response anyway. What the development of civilization clearly shows is that the use of ratios and proportions and portions was well advanced by the time any cities are evidenced, and especially in Egypt but probably in the Indus valley region first the use of a simple tool called a ruler conferred great economic and cultural power on individuals who could master it. Howevere this did not preclude having to coerce others to ones ways and educating ones offspring in those ways to ensure power remained with ones family and tribe.

I have to write this as i know from experience that the bon mot does not last with me long.
 It is clear to me that my culture and language have obscured this notion: the set notFS whatever it is is undeniably an iterative space, a recursive space if you will, that is a repetitive space and  a periodic one which is active at all scales and self similar at all scales.it is a space of generalised boundaries that surround generalised regions in which periodic and cyclical change occurs and from which periodic and cyclical rotation arises; that is it is a space of motion; pure motion that is vorticular motion and  vorticular at every scale and in every boundarised region. That is it s the universal fractal generator of a fractal spaciometry and a fractal self awareness which i call the experiential continuum, my experiential continuum.

Thus a fractal formula that is iterative, rotational in a vorticular motion and self similar will be a necessary description of the elements of the set FS and should provide an approximation to the activity in the set notFS as we iteratively uncover it. Such a formula as an approximation will be achieve through a Taylor expansion of the Einstein field equation of general relativity and special relativity such that the additional terms ar identifiable with the Dirac equation and the Feynman field computation schemes common in quantum theory. The additional terms will be subject to Clifford algebras and symmetries and will formally be equivalent to adding a spin term to the Einstein field equations, but a relativistic spin component that reflects the fourier nature of the vorticular motion. in quaternion space.
 Progress should be made though the taylor expansion of the tensor formalism, and a modification of the harmonic oscillation model to a fourier modeling of a vorticular motion field with a lattice type structure initially that is relativistically in motion as if changing into a liquid or gas phase.
 

What this means for string theory is that the strings need o be open and in fact parts of of a spiral. The strings are not closed but do get close to closing through the spiral completing a revolution but the pitch of the spiral may be quantised but small, so that it overaps as close as possible. The second condition that determines this will be to replace the harmonic oscillations with fourier modeled vorticular motion of the strings. Thus the pitch of the spiral segment will allow for the maximum amplitude of the vorticular string. A closed string will represent a special mode that is probably stable like absolute zero or something like that, or possibly even the vacuum stable state for "zero point energy", the moment of the big bang!

The other point for electromagnetic theory is to do with the electomagnetic wave in fact any wave. We are used to waves being harmonic oscillations of a string and phase shifting in the waves, but i have yet to hear of a vorticular wave transport along a string as a model of an electromagnetic wave giving the pitch of the wave vortex as half a wavelength.

This would give a conical wave transport for radiation and a helical wave transport for lasing and masing. Focusing and refraction and diffraction and interference will need slight modification but may explain why light is bent when it goes through a slit in a barrier. The interaction of light as  "particle" with a slit does not seem explained by saying it is just bent. How is it "bent",do the particles bounce around in the slit channel? Is that how they spread out when they leave the split? When a wave transfers through a slit it oscillates the space in some way so the slit becomes a wave source, and the resultant wave source emits the transferred light wave as a coupled "light resonator".

So now i venture to state that the laws of thermodynamics are incomplete.  I venture to say this because i have sufficient comprehension of what i am about to notice that this is an obvious missing area in our understanding of notFS.I do not claim to have full apprehension of the details only to comprehend that details are missing. It is obvious to me that the increasing "order" in living biological and zoological systems flies in the face of the thermodynamic description of relativistic motion. I might add that now i am encountering "fellow souls"  using this term that i use it in its relational sense and not in the sense of being  coherent with Einsteins postulates in his theory of special and general relativity.

The vorticular nature of notFS is only partially described by the current expression of thermodynamic laws. It seems clear from the Dirac analysis and the Feynman analysis that a symmetry has to be expected and looked for in all things. Thus when we look at radiation into the vacuum we need to look for a radiation from the vacuum into the region that is the focus of our current interest. The uncertainty principle of Heisenberg means that in effect by focusing on one aspect we exclude other aspects, obvious i am sure when stated in these terms but nevertheless impactful on our theory making.

There is therefore to be expected a thermodynamic principle of increasing order with decreasing relativistic motion. That is to say quite naturally that as relativistic motion decreases in a region relativistic order increases in that region and vice versa. Clearly relativistic motion should be modifed through any rules for relativistic motion transfer, and such rules need to be symmetrical allowing transfer both ways. This is where i think thermodynamic laws have missed something.

We already have this relationship in phase states and phase state transformations of matter so it is not hard to point to examples highlighting the missing details in the thermodynamic description of notFS.

As a vortex spirals in it will also spiral out, but each event in sequence and in relation to boundary conditions. Thus a vortex may spiral to a "point region" as in translating to the "point" and then spiral out of the "point" either continuing the translation or reflecting or diffracting the translation depending on the boundary conditions and the "point region" permeability properties to the relativistic vortex motion. Alternatively the "point" may remain where it is but increase in relativistic potential motion, and or relativistic spin motion, that is rotational relativistic motion transfer may take place to the "point". The "point " thus may appear to grow in some ratio/tensor sense or intensify in again some ratio/tensor sense. So as examples it may increase in spaciometric mass or in mechanical mass, or it may increase in spaciometric density or mechanical density. or it may increase in spaciometric rotation( yet to be defined but it is on its way!) or mechanical rotation. Contrariwise it may appear to shrink or diminish in intensity if the vortex spirals out of the point.

So what am i to say about the fact that vortices usually have a low pressure area at their core in fluid dynamics? Well it could be as simple as saying the vortex is spiraling out from the low pressure region  in some sense, which is counter intuitive i know but interesting nonetheless.

Spaciometric rotation I will define as tha part of the boundarisarion process that apprehends the boundary of a form.

 I have come at it from this direction because rotation is so intuitive that the issue of defining it seems unnecessary but it is now necessary as I have a background theory that is the Logos response against which it is to be compared. Nevertheless it is an axiomatic notion because the sense of rotation is in fact located in the ears. The audiometric sensors are fascinating and instructive in so many ways, but suffice to say at this time the rotation sense is a synesthesia of tha audiometric and the visual and the  Gustatory sense systems.

So what is boundarisarion? I perceive in the visual and kinaesthetic and auditory sense systems a form and variation in ratios within that form. The form itself has a property called relatedness or more formally relativistic motion. This relativistic motion has a distinction in the sensory measurement schema or procedures which I identify as boundaries. The iterative measurement procedures provide me with sufficient ratio information to be able to perceive a boundary between regions of ratios and to perceive regions via the sensory "arrays". By identifying these perceptions of distinctions I provide myself with a recognition procedure that is iterative and foundational to subsequent iterative processes and schema of perception at higher or more extensive levels.

At the level of distinguishing a form therefore I am able to perceive a global boundary or rather a spaciometric boundary as we ain't doing just geometry here! It is thus interesting to note the levels of perception of boundary as we alter attention and or focus on the form. It is curios to observe the flood of boundary ratio information that becomes available to perception as we do so. This is tantamount to the iterative processes going on more or less at an unconscious level, providing the CNS with continuous samplings of measurement data vis a vis the form I am focussing on..

Thus I am able to perceive the form, the surfaces the edges and the corners and the relative structural components of the form as a whole and it is this whole perception that has an inherent boundary Associated with it. This boundary is perceived but not apprehended and thus not comprehended initially. To apprehend a boundary I have to trace the boundary of the form. This tracing of the boundary is not just by pencil and tissue paper! It is in fact recognizing the perceived boundary and it's structure which will consist in parts of surfaces edges vertices or corners . These identified names represent a cultural decomposition of form at the boundary structure. Tracing the boundary then is tracing and identifying the forms surfaces structures and relationships in an entire form. This is spaciometric rotation.

It is clear then that spaciometric rotation is complex and is an activity that encompasses the whole form and it's parts. Because of this the kinaesthetic sensors are involved in the concept of spaciometric rotation  too,because I literally have to act to perform spaciometric rotation, whether it is just movement of the eyes or a movement of my entire body the kinaesthesia is engaged in tracing the boundary.

So as simple as it may be rotation is a fundamental whole organism notion which deserves to be axiomatic, but which reveals a richer understanding of my interaction with notFS when I do explore it.

NotFS by this definition has a structure of rotations within rotations at every scale and within every boundary. In addition the rotation clearly does not exclude extension or translational motion or relativistic motion.

There is a fundamental realisation now apparent in spaciometry which is that boundarisation is a fundamental element of the topic. I have mentioned that rotation does not exclude translation, but it is now obvious that every translation is bounded by a rotation. Thus there are these fundamental motions in the topic of spaciometry : rotation and extension but every extension is a part of a spaciometric rotation. This is not obvious if one uses an abstract entity like a Euclidean line but in spaciometry such things are abstract tools. Thus any real spaciometric form is bounded and therefore within a spaciometric rotation. The idea is useful in that motion  especially relativistic motion does not then have to be perceived as motionless just bounded. Thus relativistic motion forms that is relativistic forms are bounded by a rotational motion which I have called a boundary. When I say rotation here I am saying spaciometric rotation, but the analogous idea needs to be explored by myself to find out how much this is already in use in mathematical activity. For example I can already see a connection to the Heine Boreal covering theorem,and certain discs and spheres used in Hausdorfs dimension theorems.
which leads me to suggest that  a serviceable theory of everything formula will naturally arise from a reference framework that emphasises rotation and extension. Such a non cartesian framework already exists and i think to pursue its development in spaciometric terms and in relativistic motion terms may be useful to my quest for the foundations of mathematics.

 i probably need to look at clifford algebras as well, as i have a rough formulation of a relativistid reference framework based on binary heaps and pointer notation.

I want to define a spaciometric locus or path that is a "curve" in a general sense not specified here.

The locus is a path that apprehends every boundary of a spaciometric form in sequential manner. this ;ocus will therefore by definition be a complex apaciometric rotation in general. I will call this a spaciometric vortex locus.

Some examples of a spaciometric vortex locus are a spiral and in particular a vorticular spiral.

While a vorticular spiral can encompass a simple form it may not deal with more complex shapes like trees etc. A combination of spiral vortices  may be needed to proscribe a complex form.

However a peano curve may be a special example of a vorticular spiral that can deal with complex spaciometric forms. Now i can define a length along a spaciometric spiral that proscribes a form as a spaciometric angle measure.of that form.

Thus a spaciometric spiral will be my tool for defining an angle measure in spaciometry where it makes sense, and i will take whatever element of a spaciometric  spiral necessary to make a useful set of distinctions and ratios.

There is a particular spiral form that I will adapt for angle measure. And that is
P(r,r), in which the polar form indicates the tensor is ( Angle measure in length( a) around unit circle, radial length(r)) and a=r. This is a spaciometric formulation so it applies to so called 3d forms and surfaces not just to 2d surfaces.

This spiral is quite interesting and a challenge to construct without an existing reference framework. This highlights that every curve or formula is dependent on the referencing framework, and such a framework must exist or be constructed before it is used to reference from. So i have to use an existing framework to describe the curve that will be the reference framework!

I can use the standard polar coordinate framework or a modified one. I can use a unit circle  as an angle measure in radians from which i can ratio the radial length of the spirals locus from the origin. However to apply it to a spacometric  vorticular spiral i will need to use a second and third angle of orientation of the radial length to pick out regions on the spaciometric form P
(ø,Ω,¥,r). When this spaciometric spiral is determined, the points can be referenced by a spiral length ξ . The construction framework serves as a translation framework back to cartesian or polar coordinates.
 The spiral framework has to be fractally structured to cover the whole of the space as well as fractally divided along its spiral lengths to provide measurement criteria. This is truly a relativistic reference framework, relative to the form or forms that this vorticular  spiral ratios.

An orientation sphere has traditionally been the angle measure reference form, so a vorticlar spiral as an angle measure changes a whole lot!

I would attempt to trial this system with a slinky to see what it reveals.

So it becomes apparent that using a vorticular spiral locus as a tool to reference frame a spaciometric form is only the beginning. A vorticular spiral is an abstract tool, which in all instances would lead to an infinite length in being applied to a spaciometric form. In practice i wold only use a finite lengthed vorticula spiral to reference frame a spaciometric form. But now it seemed to me that i was attempting to "wrap" the vorticular spiral around the form and i already know how to do this because wrapping spaciometric forms is a common activity, especially at christmas and birthdays!

So i realised that i could use a finite spaciometric surface to wrap a spaciometric form following a vorticular spiral locus. This wrapping would cover more of the spaciometic forms surfaces in a finite  lengthed  vorticular spiral  and would be called a vorticular spiral wrap.

Thus i defined the process of applying a finite vorticular spiral to a spaciometric form as wrapping the form;  the surface that is used to wrap the form as a vorticula spiral wrap; and the adjustment of the surface dimensions to achieve the best wrap as a process of optimising the wrap, that is finding the minimum finite dimensions for the maximum finite surface area of the form.

This is a bit like Dirac's instance integral in that the vorticular spiral locus wrapping the form will be the limit case of an optimisation process which may always have the same known value.

Now the theorem would be that every spaciometric form has a finite vorticular wrapping for each relativistic point of view.

It would then be nice to show a uniqeness of an optimised vorticular wrap for a relativistic view and a fixed vorticular spiral locus.

Now the sweet thing: a vorticular spiral locus and thus a vorticular spiral wrap is a relativistic reference framework for the form. This relativistic reference framework could totally represent the form. and in general all spaciometric forms could then be represented by vorticular spiral wraps.

This then means that any general equation is only dealing with one spaciometric structure for all forms in spaciometry: a vorticular spiral wrap.

Once the properties of vorticular spiral wraps are explored i imagine that they could be easily applied to all scales from the quantum to the classical, and would represent quantum characteristics at small scales because P(r,r/n) would characterise small wraps and P(r,n*r) would characterise large vorticular wraps where n is an increasing rational number.

Now i had some difficulty in visualising the construction of a vorticular spiral because i wanted to refer to a spaciometric form that was nascent not preloaded with presuppositions like the circle form does for angle measure. I know that i cannot stray to far from the trigonometric ratios in their fundamental descriptive power, but i did not want them to be apriori to the vorticular spiral. The solution to this for me is the use of the cone shape as an apriori measure just as the orientation sphere is the apriori measure for all orientation and angle. I will call it the spaciometric orientation cone from which vorticular spiral wraps may be determined. Of course cones are tangential to interior spheres the tangent forming a circle which lmplies trigonometric ratios are inherent in the cone measure, and also extends to ellipsoidal measures of ratio for vorticular spiral maps. The full conic sectional resources are accessible through this measure.

Field equations would then only have to be developed for vorticular spiral wraps to describe relativistic motion and relativistic motion transfer at any scale within any vorticular spiral reference frame, and the self similarity within the reference frame will be built in along with the iterative spiral process which to me means that the equations will produce and capture the  truly fractal nature of all things.

i posited once that all motion could be understood as traveling wave motion, and that inertia would relate to the establishing of a standing wave in an object to which momentum was being transferred, kind of like a latent heat function found in phase change scenarios. I did not know what i was talking about then and i do not know what i am talking about now  :D but i hope someone else does and finds it useful.  ;D

OMG vorticular spiral wraps are everywhere! :worm:

One point in passing Using a unit sphere inside of a cone enables the vorticular spiral locus of a cone to be drawn from the cone apex, and the spiral length becomes the cone s spaciometric angle which using P(r,r) makes  the tangent length to the cone from the apex 2π.  :surf:

The length of the spiral once drawn will have to be calculated or measured and a rule found.

OMG cones are the business! If i draw a radius from the apex to the base of the cone and use this as a meridian i can spaciometrically rotate around the cone by : a circle, an ellipse, a parabola followed by a negative/ opposite parabola and 4 hyperbolas  each on an orthogonal plane to its intersecting neighbour. with parallel planes having the same hyperbola, and of course the spiral. Each of these rotating motions represents  periodic locii.

I can imagine these motions in the clock, the planets, the pendulum and the jointed pendulum, and in the watch spring( an old fashioned motive force) .

As to my exploration of finding the length of the spiral. A spaciometric investigation of the cylindrical helix led to a sketch of the conic curve shape, slightly different to a right angled triangle base 2π with a height that is the pitch of the triangle. Once  had identified a rough formula and a rough shape i found myself in the kitchen stirring rice pudding.

What's the best way to stir a pot? Well i have done this exercise many times over the years and am of the opinion that to stir in a circle is the least effective . My personal choice is to stir in a grid pattern. However today i had spirals and spiral wraps and vorticular spiral locii on my mind. The vorticular spiral locus into the centre of the pot and then out created a satisfying wave effect on the rice pudding surface as it heated up, but that is too much effort to do for long. i went for my simplest counter stir: semicircle, cross the pot then opposite semi circle, a figure of 8 type stir. Suddenly it reminded me of the yin yang symbol, a symbol i have already associated with vorticular motion. That created an interesting pattern on the rice pudding surface.

 Suddenly it dawned on me that this was like the triangular pattern i was looking for to measure the length of the spiral . After i dished up the rice pudding i considered the spaciometry of the yin yang symbol. it was then easy to see the spiral "comma" forms as two rotationally symmetrical nets for a cone!

The ratios were 1:2 for each net to the whole mandala and 1:1 for each net. In addition each net curled exactly up into a cone with each part of the inner S rotating into its rotational counterpart and the long tail curling in like a cats tail to finish off the  base of the cone.

I looked and i did not find this any where on the internet. So any chinese/ japanese monks who can tell me if this was already known  in their traditions would be welcome.

You see quite simply what i have is a net that gives me the spiral length simply and the two cones of the conic section models.

So now i wake up this morning and i am simply stunned! Clearly the vast computational web i exist in has further processed this information cos i sure as hell have not! I was asleep , dreaming about life and stuff. Granted i had noticed the wave form in the centre of the mandala as i played with extending the cone net to make a cone with more and more spirals, more spaciometric rotations, more vortticular spiral wraps, and yes i had noticed that as i did this i ended up with a familiar and well known spiral form. The name of it, I thought was a semicircular spiral, after the method of its construction.

Researching this now, as an aside has been fascinating and i am going to name it the yin yang spiral, because i suspect it predates Archimedes and Fermat by a significant periodic length.

So back to the story:so i wake up and suddenly see everything is connected to this form of spiral! The cones , the spiral wraps, the spiral locii, the vorticular wave concept, the conic sectional motion concepts, the space time manifold concepts, the conception of how to tackle the Dirac equation, the Feynman field calculations, the vast and unending vorticular field concept of Maxwell( which some call the aetherand propse an aether wave theory but like Einstein my preference is to cut to the chase and sto the endless searching for some "mythical" substance and just let it be space that has all these properties or more specifically attributes. This is what i mean by space being substantive, and why i am porsuing a relativistic motion description of attributes.)

So it is all boiled down to this simple description: the yin yang spiral!

Then i noticed that a problem was shown to have a possible solution: How are a transverse plane oscillation and a radiating circular wave transport linked to a spiral form? I was constructing the solution by a compass and ruler in order to play with my cone nets!

Two interlocked spirals phase shifted were what i was drawing, then cutting out to curl back onto themselves to form cones, perfect cones, and that was only the beginning of my designs.

WTF :wtf: Have i stumbled into here!?  

Then almost as if to corroborate what i was thinking i remembered some space time manifold equations for Quasz which i have not published. You look and you judge!   

http://www.fractalforums.com/index.php?action=gallery;sa=myimages;u=410

Check this out !

And in addition i have pointed to general reference frameworks being, in general, tensors. I suspect that not all tensors thus defined will be invariant under transformation, so i introduced the names only for two more types of tensor Flexors and Growors. The intention in the names are clear i hope.

According to the way i am pursuing this, i have now my standard vorticular wraps in these Spiral Tensors and the length along a spiral is the Tensor reference system which will then uniquely determine a location in space relative to the spiral form the, observer, and the conical orientation standards.

I cannot avoid the connection to DNA as a reference framework for biological structures and so i will refer to these vorticular wrap tensors as DNA tensors.

A note about my language and style.

Some of you guys and gals  may be put off, not only by the word maths, but also by the way i put words togeher, and "needlessly" complicate words instead of simplifying them. And also apparently i might be a bit of a Mr. Micawber in these tendencies. To you  i apologise.

However that is not to say i can change my style, because as i frequently point out much of what i write i do not myself fully understand. So while this is not quite mediumistic automatic writing, or dictation from spirit guides, etc :laugh: it very nearly feels like it sometimes!

I explain it as a kind of channeling from the universal computation that is going on all around us to which i contribute my part by thinking. Garbage in garbage out applies here. So if i ask the right questions my brain is able to access and download the right kind of answers and process it into a form i can understand and utilise. It just so happens that efficiency demands the most information in the shortest form which unfortunately means very concise and terse language forms with as much redundancy squeezed out as possible.

As a case in point: from my understanding or rather apprehension of english grammatical forms there should exist a word for vortex whirling motion. I decided it would be "vorticular". i had no idea if the word existed or not it suited my needs, and after all these pages, although they have an open invitation to all to comment, are in the form of a journal, a diary of notes to myself.

So i will leave it up to you to find out if the word exists or not, but i could equally as well use helix, helical, conical helix, vorticose, vorticial etc. Words are tools, however, and the particular forms i choose represent a focus of investigation. And i am investigating and exploring vortices.

If i said to you that a yin Yang is a fundamental measuring tool for particle physicists, would that seem odd to you? :rotfl:
Can you see how much fun i can have with this?!

Its a kind of 3d protractor related to the spaciometric conical orientation standard, and would be applied as a kind of vorticular wrap.

This yin yang tool would measure the cone angle and reference the associated DNA tensor. So now i got to figure how binary heap math is involved with all this.

Another use of the yin yang mandala is to describe earthquakes. If i extend the yin yang using the semicircle spiral pattern and then join the two centres of the yin yang by a line, i can translate the two parts of the diagram along the line. So simpy i have split the yin yang by translation and made the nested semi circles obvious on both sides of the line.

Sliding these two sets of semi circles past each other enables me to create different spiral patterns.

the relation to earthquakes is the circular radiating compression wave sliding past each other either side of the slipping rift.

This may mean that a vortex will produce a radiating compression with an opposing translation along a spiral diameter, or that a radiating compression and a translation along a line through the source of the compression in opposing directions will generate a vortex, or spin or "angular" momentum.

The ying yang 3d television system: first have a binocular camera system with each lense scanning 180 degrees out of phase onto a yin yang camera sensor. The arrangement of the pixels on the sensor and on the tv screen will  be yin yang

                r         g  b
               b g         r

The binocular signal would raster across the screen in opposite directions so that the shortest light path favoured the requisite eye. With tuning of the raster speed, the pixel distance for the yin yang screen and or camera sensor , and testing for epilepsy should produce interesting diffraction patterns that the visual system can interpret in 3d.


Now the paths around a cone can be recorded in a spiral wrap of a spherical object. The paths are s  p and d, and i would venture a fourth one called e.

The s is for a circular path and e is for an elliptical path. These would represent the s shells from the Schroedinger wave equation solutions.

The p is for parabolic and this would represent the p shells from the schroedinger equation solutions.

The d is for hyperbolic and represents the d shells

Finally there is the spiral which i have not found an analogue for in the solutions for the schroedinger yet but would guess at a spin solution. The spiral on the cone would project onto the sphere as a loxodrome.

 A yin yang spiral group would wrap a sphere, but orientation and spiral chirality are important on a cone, and the spaciometric vorticular spiral wrap produces a kind of inside out cone as well as a more familiar cone.

Looking closely at the physical manifestation in quasz of a so called yin yang centre is instructive. Close obsertvation shows me that while these are not spirals they ain't circles either!  My guess is that they are a set of nested near circular ellipsoids.

When i think about the idea of a vorticular wave therefore i have to pick out the conic section forms that may mimic it. So i have not yet drawn a set of nested ellipses and split them on their common diameter so as to slide them past each other, nor have i yet done so for the parabola or the hyperbola.

Now thinking about earthquakes and the compression and shear waves s and p helps me to understand that ellipsoidal waves could explain earthquake damage near the epicentre, but further away the sequence is more marked due to the time difference in arriving at a point. So while a spiral might encompass the points of destruction it is not a straightforward example of a vorticular spiral wave, but rather a testimony to the vorticular spiral framework which can describe these spatial arrangements.

The relativistic framework may possibly be represented by the tensor V(r,ζ ) where r is a tensor spanning the space, and ζ  is the spiral length to a point in the object reference by r. r references the apex of the spiral wrap- which i will now l use to refer to the general vorticular spiral wrap.

 I feel the logos response compels a spaciometry/geometry, and a geometry compels an algebra; and it is the algebra that compels a language including the so called mathematical language.

The algebra compels vocabulary, syntax. phonemes/graphemes, parsing, grammar and composition; as well as verbs and adjective/adverb reference. There are also prepositional rules and conjunctive rules guiding the use of all these elements, and of course a community of practitioners overseeing and enforcing certain cultural standards.

The fascinating thing about this all to me now is that this represents tne result of processing ratios in my CNS, neural network distributed, processing system; which iteratively samples notFS and reveals a spaciometric fractal design based on spaciometric rotation and extension.

Which seems to imply a vorticular arrangement and entrainment of all things.

So looking again at the ying yang sculpture i can now see that they are concentric rocking circles! Thus the centre disc looks like an ellipsoid because of perspective , but it also is a good approximation to the ying Yang in 3d .

http://img1.jurko.net/2554.gif

Hey  :banana: look what i have just found! http://www.geodynamics.org/cig/community/workinggroups/mc/workshops/litho2008/presentations/Tackley_MCW_Davis08.pdf/view.  chilli

Check it out. Not my spiral reference frame but inspiring to me.  :groupbutts:

Just thought that the sin and cos functions would be a helical function when extended to 3d. Then i could use Fourier 3d helical function interpolation to model vorticular wave behaviour.

http://www.youtube.com/watch?v=Qif8K7fc6uY&NR=1




                                                                         http://www.youtube.com/watch?v=LeIF1mG500o&feature=channel   

if you have epilepsy do not watch this one!

http://www.youtube.com/watch?v=6l0Ul6AgBqM


http://www.avantgravity.com/3d_gravity.html

spiral packing (http://docs.google.com/viewer?a=v&q=cache:57gbAffAqOwJ:www.lps.ens.fr/~boudaoud/Publis/Boue06Spir.pdf+3d+yin+yang+spirals&hl=en&pid=bl&srcid=ADGEESj0fviwlkluloIsDxEVZdrXtz2Go7i9gYF9xNcbSgzFM-gcHT88oKokPAr8gCzMFGPOVJMQcrjpUtiu9G01zkBgqGv4A8ZaaaE_Uo3PHAhxq7u1riQTM1PUL-82IAafgz45DVeQ&sig=AHIEtbQhi24T-gB0PrwYu7tUlRzFSTbUHw)

 :surf: Hey just surfing man! Check out the ying yang crop circles!
http://kylepounds.org/science/crop%20circles.html

Is this guy me or what? :toast: spiralman (http://www.spiralodyssey.com/)

some more good stuff  nanomagnets (http://docs.google.com/viewer?a=v&q=cache:p47Va5Pdn

20J:arxiv.org/pdf/0705.2445+3d+yin+yang+spiral&hl=en&pid=bl&srcid=ADGEESgg-5lLaSaNtr48PZWUINtfmSjiDKA8zofGGFm7MfOq3fPgfRcsZNZmbbyYM_wNwY7uW3-FXwXkTNWx7N6FbkI9cn0VGJ4Y3oj9Cm9fbazlBwULhyxwvYxf0uh8HJZakhJUPK5K&sig=AHIEtbRU9AjsyIgRreEWJ2HzDypRZh11dA)


http://vimeo.com/groups/114/videos/8979430

http://www.chinesefortunecalendar.com/YinYang.htm

the sin cos function in 3D is probably closely related to the complex log screw :)

Ha, nice :)
Just a few days ago, I compared a tennisball to Yin-Yang and stated, it's like a 3D version of it. I actually meant it as a joke but now you send this pdf. Nice.

Hiya Kram1032, yes very nice.  :drink:

I am looking at weaving now so some more good stuff.
 baskets from spiral wraps (http://www.primitiveways.com/basketry2.html)

Also there is a relationship between the Archimedian type spiral length and the euler formula which i have not worked out yet.

Just realised that if i can define a perpendicular to a spiral loci i can uniquely reference every "point " in a region by a parameter ζ the length along the spiral and a parameter x the length along a perpendicular to the spiral at that length thus reducing 3d to 2d, just like that, as Tommy Cooper used to say. so using a regular helix and extending the radius to any length x gives me roughly

ζ=ø+ø/2π*pitch and the coordinate system S(ζ,x). with x= r the radius of the helix defining the axis of the helix. Although this is not a conical helix reference frame it is a start! :D

The great thing about this "slinky" reference frame is it does not have to be straight, and it can follow the contours  of an object while it is itself being referenced by another reference frame, slinky, polar, cartesian, elliptical etc. For a relativistic reference framework i need at least two reference frameworks that are linked. They can be the same but with a transformational difference, giving at least a stereoscopic view of the tensors,flexors and growors.

Now i have to say that function is a bit of an old fashioned word and so i guess i am going to use Rule as a simple designate, and Linking as another. Then i can have linkings which are the rules if there is no simple sequential or programmable action or iteration on the elements of one parameter relating it to another parameter and its elements.

So i can still have fun linking things even if i cannot give a concise rule of what is being linked: these will be like parameter data sets.

Here is a question for cosmologists. If a vorticular spiral locus , that is a conical helix locus is used as a reference frame does that support or trash an expanding sub atomic "particle" theory of gravity?

I don't know the answer so i guess i will look at it when i get through working out the relativistic motion model i am intuiting. :embarrass:

russian spiral (http://docs.google.com/viewer?a=v&q=cache:yd-449R9psQJ:www.beadpatterncentral.com/russianspiral.pdf+spiral+as+a+weave&hl=en&pid=bl&srcid=ADGEESi3EVzsbT5vCky_bFm8YWorIVMySkkdEiY-lrpv5l3lsykKvRioBad7l17asUhaJH11kuIMdbU_ZrHVzqi-FpIYnKzG1musZOAct4Qg4AYnXe1oO7RcLci-W4BaDdpyjt68_5AG&sig=AHIEtbRQABJf0jn67R0pmQzJz2GxsG_yPg)

http://www.youtube.com/watch?v=M_sHmpagq5Y

http://suzannecooper.com/classroom/spiral.html

Just some practical examples of Links and Rules, and it is clear that these are classed as Instructions and  a particular type of Instruction would be a Recipe  the finished product is a Construction. So doesn't that sound better than "function"?   :over:  What do you think?


my favourite spiral wrap so far (http://www.weavezine.com/fall2008/wz_fa08_EricadeRuiter.php).

 Instructions (http://ezinearticles.com/?Balloon-Decorations---The-Secrets-of-Creating-a-Spiral-to-Do-Balloon-Weave-Arches---(4-of-4)&id=2659871) of a more general nature, then applied to a specific construction.

Rule guided automatic machine (http://www.demirci-chain-link-fence-machine.com/double-spiral-chain-link-fence-machine.html)  This is an automatic machine so it has an algorithm that links the elements of the parameter and produces the construction. Algorithm then is just another way of saying Rule.

So i guess i can use more accessible language to talk accurately about mathematical ideas.

There is a certain set of numbers/numerals/polynomial  power series expansions/ratios- which are  fundamentally linked to rotational motion in spaciometry. These ratios are called logarithms. These logarithms are on a curious base which is called e the Euler constant.

The relationship was first devised by John Napier (http://johnnapier.com/table_of_logarithms_002.htm) using a projection of a surd line onto a parameter line. Napier, interestingly describes these ratios in terms of motion along the parameter line. He describes the whole interaction in terms of variable speeds along the parameter line. He was indeed an admirable fellow (http://johnnapier.com/table_of_logarithms_001.htm)!  :-*

So now the laws of thermodynamics are for increasing entropy systems but what laws are for decreasing entropy systems ?

I will call them electrodynamic laws, and these laws will govern decreasing entropy and self orgamisation and self asembly arrangements.

So a little background musing to illuminate my thought path.

So i take the hypothetical electron and ask does the hypothetical electron travel in a straight line? So i accept that it may be good enough to think of it doing this in a vacuum, but i do not accept it as the accurate description. soo now taking it as moving in some straight line what happens when it encounters a dense substance? does it travel in a straight line? Well no, if we use the rutherford model of the atom and the molecular structure of compounds it is accepted that the electron interacts with the substance by either passing through the "space" between the atomic nucleus and the electron shells or being deflected by the electron shells or joining a particular electron shell around a particular atom. so the fate of the electron is to be interactive with the existing molecular structure and so does not travel in a straight line.

The next question is does the electron that goes in even ever come out? Well again the accepted idea is in a lattice structure the electrons form a kind of cloud around the nuclei and this is how electricity flows through certain conductors. Well a cloud is bot a good analogy as the flow appears to be incompressible, so a fluid analogy seems more appropriate, and the principle seems to be that if an electron goes in one end an electron comes out the other end of the conductor. All pretty much like plumbing,except that the electron that come out is unlikely to be the electron that went in, because of interaction.

We see this more clearly when i look at a liqud structure supporting a current flow through an applied potential difference as the flow is now described as ionic and as a fluid flow of ions. So now is the electron that goes in the electron that comes out as we are ususally shown? Well no because the chemist will point out the chemical interation among the ions that promote the formation of certain compounds in solution. even some of the compounds may precipitate forming a more solid but amaorphous structure that settles to the bottom of the fluid away from the action potential.

In a case like this it is even less likely that the electron that goes in ever comes out,

But what we do have is a catlysis of a reaction by an electric potential.

Now i jump to an enzyme whose job is to provide substrates with a place that lowers the action potential for their reaction . thus i see that a particular molecular structure is in fact acting like an electric potential difference faciitating specificc kinds of reactions.
i then jump to these enzyme "batteries" having a specific electro potential that varies around the shape of the molecula r enzyme whic encourages certan other electro potentials to either bond or deflect from certain sites on the molecular structure. That whwn the right potentials bond with the right sites the battery circuit is complete and an actio potential as a current of ions flows between the two parts eithe r bonding or breaking bonds. The product then falls a way from the site of biding because it develops a repulsive or non attractive electric potential. If the bonding site is open a neutral electical potential will be easily pushed out of the way. If the site is sunken a repulsive potential may be needed or a change in the molecular arrangement that expels it. For example a potential may cause the molecule to coil like a spring which then returns to its normal state expelling a neutral potential.

So in this case the electron that comes in i involved in and mediates chemical reactions on a molecular template called an enzyme. Such enzymes if they are highly iterative will confer a self arrangement onto the substrates and thus induce self arrangement due to electrodynamical resonance modes solely.

Thus i have an inkling of hoe electrodynamic resonance may lead to self assembly and self organisation which is an decrease in entropy.

So now if i run through this again but use quantum dynamic descriptions of electron transport i think it is obvious that the electron that goes in is unikely to be the electron that coemes out, but an electron comes out in an electron transport system whatever the phase of matter , and this weaving of the electrons is what creates order.

Now if the electron is actually a vorticular wave?

 to be corrected

Check out this cone.
This is how Yh¶h does vorticular wraps:
(http://www.fractalforums.com/gallery/3/410_27_07_10_7_31_58_1.jpeg)   note the loxodromes start from a ring not a point.


This is  a picture of an atomic nucleus made up of vortices:   

                                                         (http://www.fractalforums.com/gallery/3/410_27_07_10_7_31_58_2.jpeg)

This is also a snapshot of a vorticular wave:

(http://www.fractalforums.com/gallery/3/410_27_07_10_7_31_58_0.jpeg).

Pretty neat for a pine cone huh!

Now one particular vortex fantasy i have just realised is the  translational speed fantasy. Vortices do not create terrific  translational speed as the object nears the vortex centre. A little drawing showed me that a region following a spira path undergoes spaciometric rotation, and that this rotation accelerates the closer the region gets to the spiral centre. Quite simply a spiral or vortex path is an accelerator of rotational motion, and there is no upper bound to the spaciometric rotation rate of change, as following a spiral path accelerates by definition.

with this in mind i move to an explanation of solenoid magnetism. The effect of a spiral is to generate spaciometric rotation in regions that follow that path. Now magnetism is experimentally proportional to electron spin alignment. So if electron spin is aligned magnetic force is increased. The helical wrap of a coil entrains all the electrons to spin in the same direction at the same rate, simply as a consequence of spaciometric rotation. The electrons that flow around the conductor interacting with the atomic structure   gain a constant rate of spin spaciometrically, in the same vorticular direction. Thus a vorticular resonance is set up which encourages a vorticular coherence, that is a vortex spirals out in the helical direction. This effect is the basis of masing and lasing. Thus the usual magnetic force line diagram is a vortex cross section.

Now when a suitable material is placed in the vortex field its free electrons are entrained in the spaciometric rotation  enhancing the rotation spin density and alignment thus extending the vortex. The free electrons interact with the atomic structure so that atomic spin is also enhanced so diamagnetic effects are also enhanced.

Tighter coils should increase spaciometric rotation, but a conical helix wrap should provide the greatest acceleration.

Now in the light of this an alternating current will develop a vortex  which will collapse and then be established in the opposite direction . This alternating vortex field has its own effects on neighbouring regions.

The Rodin coil wrap has 2 spaciometric rotation effects that oppose each other. Thus the electron spins are aligned and cohered but the spin is accelerated and decelerated regularly thus generating a variable vortex field that is forming and collapsing, thus a potential null vortex field. But in addition to the spaciometric rotation being canceled by the wrap the pulsed signal resting on the 3rd cycle skews the vortex round the torus and accentuates the central vortex which is a distinct punch in a direction determined by the spaciometry of the wrap. It seems mono polar but in fact it is coherent field formation and collapse in resonance mode.

I think that one point of view is that multiplication is repeated addition, but that is a powerful but inadequate description of multiplication, because of the primes and because of cell division. You see the primes seem to indicate there is a structure to bundling regions together, and every so often one has to start a new thread, so to speak which i will call a prime thread. The ratio description of notFS admits of equivalence classes which are thread like. By not cloaking the ratio aspect of notFS and my logos response i hope to explore this structure.

The bundling together process, is an aggregation, but the division process is a disintegration, and both multiply but in differing ways structurally. Ratio analysis may indicate how, but essentially disintegration requires exponent bases to describe, while aggregation requires mod or clock arithmetic bases to describe.

So we need ratios written in fraction form, and then bundles of fractions are soon measured fractally in exponential denominator schemes 1/2^n ,1/3^n,1/5^n,1/7^n.. all prime number numerators.


Similarly we have used mod(1), mod (2),mod(3),mod(6), mod(7), mod(eight), mod(10), mod(12), mod(16),mod(20) mod(22),mod (24)mod (28),mod(29),mod(30) and mod(31),mod(220) mod(52),mod(365), mod(1760),etc to name but a few.

Currently i aggregate time under a mod (60)+ mod (60)+ mod(24)+ mod (7 )+mod(29,30,31)+ mod( 12) system .

So the structure of my aggregation is described by the collection of clock arithmetics used to specify a bundle.

The decimal system itself is a mod(10^n) bundle aggregation with a collection rule that orders them and sequences them into a polynomial series .

This way i can construct an arbitrary  mod(n) arithmetic bundle collection  and define a rule that orders and sequences them and then use them as an aggregation scheme .

Polynomials then are an aggregation scheme based on logarithmically varying bundles like mod(10^n) of arbitrary fixed base called x.

So aggregation schemes based on bundle size x are mod(x^n) clock arithmetics with a collection rule that orders the series logarithmically , and sequences them irrespective of the surface they are written on, or the region the bundles are collected in.

These have traditionally been called polynomials and used in all sorts of ways to describe aggregates generally.

An interesting polynomial system for me would be a mod(e^n) bundle clock arithmetic collection with a logarithmic power order rule, and a normal sequence pattern.  

Hamilton sucessfully developed the algebra of couples and that of quadruples called quaternions. The algebra of triples will be found in the geometry of triples as per the logos response.

Clearly the geometry of triples has been a well worked field in terms of the Euclidian geometry, and this may be why it has appeared so elusive, in that it lies obscured in everyday commonplaces. For example trigonometry must play a significant part in such an algebra, and to the extent Hamilton used trigonometry in his algebra of couples is the possible obscuring of an algebra of triples.

(60)+ mod (60)+ mod(24)+ mod (7 )+mod(29,30,31)+ mod( 12)

shouldn't it be mod((28,29),30,31)?

Hiya Kram1032, of course it should, and thanks for contributing.

I am going to suggest a relationship in the primes resembling the spiral threads of a phylotaxis. Now i have seen one and been intrigued but i do not know what it relates to beyond naturally occurring biological growth patterns.

To understand a prime i think i have to steer clear of the notions of number and stay with the spaciometric precursors. Thus the issues relate to a phylotaxis, and how to multiply, or bundle efficiently to fill a region,

Of course from observation this seems to be a spiral, and that is the spiral of the phylotaxis.

One spiral thread cannot cover or fill every part of a region and so there is a space every so often to start another thread. I think that these spaces will exactly match the prime number patterns, and may be got at by using an exponential aggregation clock arithmetic based on the known primes, and extended by every new prime found.

That is a mod(p(ø)^n) clock arithmetic aggregation where p(1)is the first prime and p(r) is the rth prime and the aggregation is sequenced by the logarithm of the clock arithmetic and the prime indicator ø. That is a 2 parameter  sequencing which spaciometrically should cover any given surface area in a kind of spiral pattern. Thus i would look for a prime on a spiral linked spaciometrically to a phylotaxis.

In my previous post i loosel used+ as a sequencer and a connective, not at all as an operation or even a set addition property.

The rigorous notion that i am using is in fact subset inclusion, but with a rule that retains exclusivity in set operations . The rule is precisely the mod(n) clock arithmetic.

So my 60 second bundle is a subset of a 60 minute bundle, but the enclosing set can only operate on complete 60 second bundles. In this way the enclosing set has a mod(60^2) clock arithmetic, but its operations are restricted to a mod (60) arithmetic on the mod (60) bundles. similarly the mod (24) clock arithmetic is actionable only on mod (60^2) bundles  included within it etc.

The interesting thing to me is that if i draw a graph of these relationships or a set diagram, i spaciometrically have an image of a shattered or disintegrating whole! This of course takes me right back to the notion of disintegration and a resonance with fractions .

It is immediately clear that whereas disintegration breaks a whole spaciometric mass into chaotic pieces in general, aggregation in general puts these chaotic piecees back together into a whole spaciometric mass, and the sequence i do that in determines my aggregate clock arithmetic system, or is fixed by imposing such a system. Pn a similar way the way a spaciometric mass disintegrates can inform me as to an appropriate aggregate clock arithmetic of fractions to describe it with, or the imposed aggregate system determines how i describe the way a thing is disintegrating.

The less i impose  and the more i follow the lead of the spaciometric form i play with the more intuitive is my understanding of the behaviours of the spaciometric forms under aggregation and disintegration.

The polynomial aggregate mod(e^n) clock arithmetic by implication should be more in tune with spaciometric behaviours than the decimal system. It would use 0,1,2,e as clock arithmetic elements where e would be the mod divisor/ signal to go to the enclosing set and i would in fact generalise to the rationals as clock arithmetic elements and would find that surds naturally would be necessary. Surds are the basis of the polynomial numerals we call complex and quaternion and indeed hypercomplex, they are also the basis of the irrationals: yes mathematicians are and were slowly going Mad! :crazy: :whistle: :no:  

However spaciometrically these ratios are very real for mod(e^n) clock arithmetic aggregates, and will include that favourite the golden mean ratio or ø.

i hope to see and show how √-1 is a departure from extension in spaciometry to spaciometric rotation, eventually. However i need a scaffolding to support the restructuring i am exploring and one that mathematicians have used in the past intuitively, and i think this scaffolding is the general polynomial form. Hence, why i refer to them as polynomial numerals

In terms of representing a generalised mod(e^n) the sin and cos ratios on the unit circle are going to be useful, whereas the tangent ratio may have some bearing on the spaciometric arrangement of the prime bundle threads as ratio threads.

And i have just developed this linking rule for the ingredients u,x,y in my parametrise recipe for a helical spiral cake, Its not finished yet

x=u*cos(.5)*sin(u), y=u*sin(.5)*cos(u) : cartesian

ø=u*cos(.5)*sin(u),r=u*sin(.5)*cos(u)  :polar.  < This one is interesting because the finished image has the yin yang in it, but   
                                                                    this is a perception as the path traced is a cardioid,

u is the parameter for the length along the spiral coil, and the trig ratios model the pitch and the periodicity.

So i was thinking: Hausdorf and his partially ordered sets  :wtf: . Spaciometrically an aggregation is either ordered by imposition or it has its own relational order based on contiguity. In general i cannot perceive contiguous relational order sequentially, it is just perceived. Auditorially i can experience this relational order as a chord for example, which is a interference pattern relationship with phase distinctions as well. This is still sequentially experienced in a sampling mode that picks up the variations and assigns additional values such as distance to source location and speed and pressure etc. this visual relational order assigns relational structure distances, spaciometric rotations. and spaciometric mass and density. The kinesthesia adds more in terms of mechanical relationships.

So partial ordering means what exactly? Is it a partially imposed order? It can not really be anything else can it as a relational order is a perception that is nascent . t b c

Yesterday was a good day. Spiralman pointed me to  this book by Ginzburg (http://www.amazon.com/Prime-Elements-Ordinary-Matter-Energy/dp/0967143241), Terry W.Gintz RSK (http://www.mysticfractal.com/rsk/rsk.html) gallery is freakin awesome and i found this 3d calculator (http://www.spacetime.us/) on my ipad apps site.

I can honestly say i have never heard of Ginzburg before now, but his ideas are not new just popularized and worked out in detail. He is in fact heir to much eastern european thinking about an aether, or a substratum of a fluidic type analogy. I do not propose to read his books yet as i am happy that i do not have to do the math so i can play around with the notions i have set out and have fun. Any way i have no one to convince but myself, and that is how it should be. I mean i do not know if what i am playing with is going to harm anyone because i do not know what i am playing with exactly. That is why it is fun. One day like Oppenheimer and Feynman i might be coraled into making a bomb or something! Yh¶h forbid!

Try as i might i find no relationship between Vitaly and Vladimir both new to me. However vitaly is a nobel prize winning physicist and vladimir is a metallurgist who has studied theoretical physics as a hobby, not unlike myself really.

Vitaly i am drawn to by the mathematics, Vladimir i will have to look at in the cold light of day to see if he is an armchair pundit, like me! :embarrass:

I am looking at packing and stacking, and more general arranging and heaping under the notion of bundling, in a motion field which has the conic sectional curve motion properties/ attributes (including helical) as a gravitational description, or law.

I have also been looking at spaciometric restrictions on bundling in cases of relational arrangements like plant branching analogies.

The bundling notion is in fact  a nascent multiplication notion, but it applies to an aggregation notion as well as to a disintegration notion. Disintegration is a special form of disaggregation, the "reverse" of aggregation.

Aggregation is a nominalisation  of a process: to aggregate, with clear links to : to collect so to distinguish 'an aggregation'(collection) from aggregation i will use the word collection or a collection.

Clearly the speed and force of these processes has not been a consideration at this stage but these qualities are quantative measures of the motion field, and are present in some mechanical and biomechanical attributes in the description,for example :stability and growth.

Aggregation consequently can be imposed by a biological mechanism( including human interaction) or arrived at by dynamical mechanisms in the environment including electromagnetic and thermal dynamic processes .

I therefore distinguish organic aggregation and disintegration, and inorganic aggregation and disintegration. This is done primarily spaciometrically so tht by inspection and exploration i can make these categorizations. The spaciometry of the inorganic and organic categories is significantly different and therefore will lead to different algebras and thus different maths. It is therefore significant that there is a deep self similarity with regard to the construction of the numeral systems. This may simply lie in the cybernetics of the observer and its sensor system arrangements.

Soo ! i was thinking about my Instruction anaology for mathematical functions and procedures, and my ingredients were parameter elements? so what did i mean by that? Does that make a parameter a set of elements?

A parameter is a dimensional fractal quantifier, that is a fractal employed as a standard reference for measurement or quantification purposes. Spaciometrically things are quantifiable but the measure is totally relativistic, that is dependent on the person, the standard, the temperature and pressure and location relative to other regions etc.

So the parameter is the ingredient and it is a fractal withe named distinctions. The named distinctions are not elements as in set theory, although there is a one to one correspondence with the elements in an abstract set like the set .

The abstract sets of number theory treat the elements as reified mathematical objects while procedurally instructing their use according to the circumstance, for example as marks along a number line, as a counted out quantifying and or ordering process (the cultural iteration +1) and often as both or flitting between the two. Culturally the rule is our only visual referent and the count our only auditory referent. This is why mathematics and music seem so intimately linked.

So what do i mean by dimensional? Merely that each use of the parameter is strictly related to a measurement of a dimension of a spaciometric form I have opined before on the confusion between this use of the word dimension as in dimensional analysis of physical quantities and the more science fiction use of the term! Dimensions derive from the practice of categorizing process of measurement and distinguishing it from a different process of measurement. These differences can be as simple as orientation differences to spaciometric form and mass and density differences. Thus the spaciometry and the cone of orientation and my own sensory sampling systems and procedures are what underlie a dimension. I therefore find it hard to subscribe to "other " dimensional objects except in the sense that the parametrisation of these spaciometric forms requires more than the standard 2/3 dimension description.

It is often hard to see that xyz are as much parameters as r and ø,as are any other parameter in maths. For example the angle is a parameter of the ratios sin cos tan. i can call them an Instruction using the ingredient "the angle" but it has to be in a right angled triangle. So the specific ingredients in my sin cos tan cake would be a whole bunch of juicy right angled triangles. mmmmm! Delicious!

i want to introduce two opposing characters into the plot line. The first is Thermal, and he always applies the laws of thermo dynamics while the second is Electral and she always applies the laws of elecromagnetic hydrodynamics.

Electral is always struggling to develop and maintain order while Thermal is just happy to create chaos! He is the proverbial chaos monster!

Electral uses all in her power from the mighty electro magnetic gamma ray to the tiniest electron spin to create wonderful and delicate arrangements of order with space as her medium.Thermal however takes space and kicks it about all over the place, giving it terrific velocites and vortices of fantastic rotational velocities. When they clash-sparks fly, thunder rolls and heat and light radiate everywhere, and space warps wondrously!

The plot thickens!!!.

I have learned today that the whirlwind (http://www.bookrags.com/eb/whirlwind-eb/) has a spaciometry with a beginning a middle and an end. So the vortex is itself part of an inclusive system which is dynamically balanced. This means that vortices can exist as distinct objects within this spaciometry or they can combine their spaciometry to achieve a greater or lesser vortex spaciometry. Thus fractal arrangements of variable interactive complexity can exist within a super vortex spaciometry.

Such an inclusive spaciometry is applicable to all scales.

It is interesting that the low pressure core is a helical vortex updraft, as an exploration i did based on assumptions that motion is not in a straight line unless that line represents a high frequency helical transport, predicted that "heat" plumes would "rise" in a high frequency coherent column vortex due to high frequency "electron" emission not due to bouyancy effects, that is lighter density rises while denser material rushes in to fill the so called vacuum .

These helical vortex emissions would interact with less motile space and establish a radiative wave transport which would eventually dissipate the motion. However if the helical coherent motion is at the resonant frequency for the surrounding systems then extended and conical vorticular flow should develop protecting the helical core. Thus the helical core would dissipate but be replaced by an updraft fed by inflowing vorticular air which when it reaches the eye of the vortex will have a fixed rotational motion transfer value, which will be dynamically balanced by a downward back pressure vortex. The updraft air would then rise under shear tension and dissipate when the equilibrium could not be maintained with a twist out at the top of the column.

Vladimir B Ginzburg is a heavy metal dude! He has produced some serious stuff. :stfu:

His tables speak of the underlying spaciometric symmetry in the universe, but this time using a spiral form as a tool. Unfortunately for me he has not gone on to develop a spiral reference frame so i do not see in his work the expected simplification in notation. So i am still living in hopes. :dink:

See attachments of his serious metal (http://books.google.com/books?id=dXiE_embYrgC&pg=PA340&lpg=PA340&dq=toryx&source=bl&ots=sT59LkH1aq&sig=bVvXYchZTtY8ZI61crtYYhBxk6k&hl=en&ei=DgtWTIjzDJGM0gTi-oX1Ag&sa=X&oi=book_result&ct=result&resnum=7&ved=0CCwQ6AEwBjgU#v=onepage&q=toryx&f=false), dudes.

Helicola (http://helicola.com/index.php?p=excerpts) a new drink that makes you see the world with fresh eyes. It also makes you pee your pants! :rotfl:

So the spaciometric form of a vortex iwas thinking: obviously the torus where the boundary conditions are the vortex itself. This has been called a minkowski bubble, but it is very like a bead used in beadwork constructions (can you see the where that may take me with a bit of thread?!) :embarrass: But i also thought of marbles-- you know how they get that unique swirl in the centre?

And finally for you boy scouts knots and knotworks. The not is interesting because it links the spiral filament trail to a general toroidal form which is not obviously a donut shape and extend the toroid to klein models for arranging vortex interactions.

Just got time to state this:

By iterating a motion statement on the complex plane  so called, and now motion statements in the Hamilton Clifford Void(for want of a better term for hypercomplex space) we have precisely discoverd the spaciometry of the vortex-torus form.

The vortex torus form is the spaciometric free form for a torus vortex arrangement without any boundary conditions other than itself. Thus the "filaments" of the vortex in this form lie on concentric torri and the centre of a torus is a spiral vortex.

If the form has boundary conditions imposed it behaves like a vortex in a fluid flow at a certain scale but like a bubble at a small scale compared to the fluid current flow.

Thus this form is a spaciometric form that will be fractaly ubiquitous under an iteration process with self referrential boundary conditions, or boundary conditions that approximate a torus.

The algebra that describes this form or rather generates this form is hypercomplex but this may change when a spiral reference frame is developed.

Spin a dollar and watch it fall. Why does it vibrate like that? The pitch increases doesn't it? but does it cut off or go ultrasonic?

Ginzburg V and B and Haramein and Rauscher they all get you thinking. But i find out from Tombe (http://docs.google.com/viewer?a=v&q=cache:ne9b5ylaqGAJ:www.zpenergy.com/downloads/electricsea.pdf+Frederick+tombe&hl=en&pid=bl&srcid=ADGEESgvSMAT3AamipLA741GSQrmkWp0HDZnW6cYoDb6UYx1ungINhLYkPvhE5zmCq5rCBoCnet-n5BMu0QWcFNMplpMwebiHVKkm09YdHthlFl_W2nBn15kUF1yuzKzLQyS7S_uqoAa&sig=AHIEtbR500urGUfv9ldZkQ2zec3n168qDQ) that Maxwell was a vortex theoretician. The trouble seems to be the aether concept. Einstein followed it for a while then abandoned it. `but not because it was wrong. I think that Einsein realised that  he did not need an aether, And rather than spending valuable time and resources on proving that an aether existed he could simply proceed "mathematically".

I take issue with this substantiation of mathematical equationing! Feynman typically relied on the procedural and syntactical and symmetrical notions inherent in the mathematical developments of his day without ever being able to "Know" what he was calculating . That is not to say that he did not have insights, but rather he took a philosophical viewpoint that it did not matter what the referent was as long as it was consistently and accurately referred to! Hence mathematical rigour was crucial.

Dirac took the view that the electron was real enough to base his equationing on and that Einstein had neglected the negative solutions to his equations. For a while he was strictly censured, but now is vindicated. There is anti matter! Einstein to be fair was not a brilliant mathematician like Dirac or Levi, and often was helped by his wife to do the calculations. He really thought that only the positive answers made physical sense, and thus used the mathematics as a tool rather than a model. Schroedinger and Dirac set out to model the statisitical and probabilistic behaviour of an electron as a real entity. So their mathematics was a descriptive , ballistic model of a particle.

Feynman was heir to that strain of thinking but did not feel the need to come down on one side of the fence about the existence of these particles/waves, especially when that meant one might be accused of supporting an "aether" hypothesis.

Nowadays some scientists in the west openly propose an aether of sorts as the standard model is so inadequate at fully explaining everything. This seems to raise the ire of some who do not seem to realise as Einstein did that life is too short to engage in this kind of debate.

Religious and mystical people want to connect the aether to their deity, but they make a false premise in assuming that space is outside of their deity! And if that is  false premise then it means that space is their deity and we can proceed as Einstein did without recourse to an aether: space itself warps.

If however space is outside their deity then what is space and where did their deity come from to inhabit space. and more importantly who else is in that space?

Fractal geometry or rather fractal spaciometry is the only geometry that adequately deals with this question, whichever premise you accept, and one can still proceed with recourse to an aether exactly as one would without an aether, except that the aetherists will feel a certain "gnosis" about why things happen a they do.

Since the mathematics is the same but the interpretation is not, proceeding mathematically seems like an attractive option. It is not.

Mathematics by the Logos Response derives feom a spaciometry, and if one posits angels in the spaciometry angel will appear in the mathematics! The Logos response is key to eventual mathematical inspiration and insight, and one cannot be divorced from the other. Feynman's spaciometry influenced his calculations just as much as Newtons did his, and Newton was a very religious man of his time.

Newtons spaciometry was an absolute space of perfection inhabited by god in which all reference frameworks were true to god. Everyone elses reference frameworks were relative to that absolute one. The only change that Einstein made at this level was that all reference frames were equally valid. This was like saying god does not have any better handle on this universe than you or I!(if s/he exists.)

Einsteins spaciometry was the emerging geometries of Riemann in particular but other non Euclidean Geometries. This of course revealed algebras which influenced his mathemaical description of physical relationships. When Tensor maths was developed by Levi And Ricci Einstein struggled to learn this description of relational reference frameworks because it simplified is mathematical description of the quantities he was relating, and it was invariant under transformation. So he hoped he could develop a theory of everything at every scale due to scale invariance.

As it turned out tensors are only invariant under affine transformations, and Levi had a fondness for cartesian tensors which meant that the properties of polar coordinate tensors were not properly perceived until recently, and are not even fully investigated now, because complex and hypercomplex tensors are more readily accessible(clifford-Hamilton operators).

Today our spaciometry is not so easily defined, but it has to be at least hypercomplex, and many are reaching forward to a fractal geometry.

When i started to explore spaciometry it was to free my mind to look with fresh eyes over  a ploughed field. In doing so i find that the field has not been ploughed but hacked at and trampled over, with a few walled gardens of exquisite beauty.

As for my own musings i find the simplicity in spaciometric rotation and extension. Two motions in a motion field which characterise every other motion. As a result of these fundamental motions my spaciometry is vorticular.

I mentioned the spaciometry of the torus-vortex as a complete spaciometric form. This form being fractal and at all scales is found in the spinning electron, the solenoidal magnetic flux field, the electromagnetic radiation wave around a dipole aerial (http://en.wikipedia.org/wiki/Dipole_antenna), the magnetic flux field around planets with active cores and stars, the gravitiational shaping of galactic masses from single galaxies to galactic clusters, and of course the ubiquitous black holes. (http://cosmology.com/BlackHoles.html)

My definition of fractal by the way is the product of an iteration process.

It is so strange how the cultural legacy i have inherited puts blinkers on the senses. By simply changing the tool which i use to reference a direct sensory experience i can avoid the blinkers of using defined terms. Thus the appearanve of the torus-vortex forms in all these examples really does indicate an underlying unity (http://www.21stcenturysciencetech.com/articles/spring01/Electrodynamics.html). And this unity is plain to see and experience especially here in fractalforums, when 3d fractal generators can routinely sculpt these basic forms without hard mathematics getting in the way.

Spaciometric rotation and extension are the basics for fractal images as well as the experience of notFS that i have through the setFS

So the reference frame that the logos response utilises is a framework of ratios. It is not a framework that principally identifies position,but one that principally identifies relations, and ratios of those relations. It simultaneously connects every region to the whole by ratios of commensuration, comparison and distinction.

By focusing i have inherently available a whole network and nest of ratios related to each other in the region of focus, and related to the whole region of focus. I have to distinguish and chose those ratios that are most relevant to me and to my purpose. This is what relativity means and what relativistic reference frames are and how i work with them, and essentially what a tensor is, and is isolated from.

A regular pattern used as a reference frame is in fact confusing t the senses, if it is not spaciometrically dense enough to hold more than 1 structure repeated iteratively. Variation is helpful because it intimates a nest of ratios that differ without monotony.

The Logos Response i realise provides an array of ratios, a structure or an arrangement of ratios, a pattern and a patchwork relationship of ratios, a framework , a nest ,a nested arrangement or pattern of ratios.....

The adjectives and adjectival phrases can continue, to describe the notion of the Logos Response and what it provides to my experience as a fundamental output from my sensory system CNS processing structure.

The arrangement of ratios are spaciometrically nested, spaciometric rotations; with "boundary crossing" signaling differing spaciometric regional relationships; and boundarisation resulting from a processing outcome of a relational analysis: so that all ratios that form a relational group have a common boundary (providing a distinction of regions can be made). That is why monotonic regularity is so hard to boundarise if it occurs on a large scale.

All spaciometric forms have spaciometric rotations, and equally i can spaciometrically rotate around all forms either fully or partially. The cone of orientation  is a spaciometric form that can be used to orientate any spaciometric form relative to an observer; and  its apex can be positioned anywhere, but principally at the eyes for: line of sight observation, and spaciometric rotation measurement.

It becomes apparent that this rich ratio structure is a data structure best characterised as a relational database of ratios. Thus it is by earlier surmising, a tensor. The logos response provides me intuitively with a rich tensor response.

There is one aspect of the tensor which is still a mystery as i write and that is the rhythm response.

The rhythm response is ultimately related to every notion of motion speed and periodicity that i can experience, and its limiting factor is system processing "speed".

My system iteratively samples the surrounding and internal notFS part of my experience, and even if the signal input is assumed to be continuous, system output has to be sequential and discontinuous. Why?, because it is the result of a periodic iterative process.

Now that period is the defining output rhythm. This output rhythm is non other than a clock cycle, but where does it come from? How does it transform from a system of "molecular" motions to a periodic clock cycle?

If all motion curves/locii are spiral at all scales it is easy to transform this through any system into a clock cycle.

I could assume a standard helical motion at a quantum scale or even a conic helical motion. The difference would be the difference between a steady state sructure to the universe or an expanding, growing subatomic structure to the universe.

Nevertheless, this does not address the question of how a helical motion at that scale beats out its own periodic rhythm. How does a spiral "know" how to spiral? How does a spin "know" when it has spun its period length?
  
This hints at a basic form of "consciousness" in a spiral or a vortex. Which is not to say that a vortex is conscious, but rather that consciousness derives from the vortex, through this mysterious property of rhythm, that is the frequency of this spiral motion, that is the vibration of this vorticular rotational motion.

However i wish to describe it, periodicity is fundamental and mysterious, and intriguing (http://docs.google.com/viewer?a=v&q=cache:wyjyDYNf9-8J:ima.ucv.cl/semanamatematica/pdf/conferencias2009/varaujo.pdf+Lorenz+by+wiel&hl=en&pid=bl&srcid=ADGEESghgVF1-Ljx6ZBmFQGVLMIWyz9FsHgWDxsf4vY0OVMJ7K2Ji9zIUFH_SWm4VN4h2G85ErFXI5apsk9GHvuZbGZ766HMQipm-LwwmrAaxTol9PyU1oiDUASwe0AV28hZU6xy2fiK&sig=AHIEtbT8DkwNGkjfhPJ9mIR9A9Oeo3ZuAA). It exists, and it exists in the spaciometric form of a vortex at its most general, and it exists at a quantum level in its most mysterious manifestation.

 The quantum clock cycle has to be some form of vortex in space, and from these vortices all rhythm and vibration and frequency transform into all that there is including computational processing consciousness and the Logos Response.

Now when i started this post i was going to end up heading in the direction that i impose a reference framework on the logos response rich tensor by choice. We do not seem to exercise that choice as we lazily use Cartesian, Polar or Hamiltonian reference frameworks, so why would i need a spiral reference framework? What would it add?

I do not fully know but i am seeing that it may add something.

I find it instructive to look at how we impose reference frameworks on the rich logos response tensor and i will go through some of that at another time.

A question i woke up with: is the Klein bottle a topogical surface for a vortex, thus enabling a vortex to self refer for its boundary conditions?

In other words can i draw a spiral vortex filament on a Klein bottle so that it retuns to the spiral filament beginning at or near where it started?

And, can stable combinations of klein bottles be formed? Or can a Klein bottle link to another Klein bottle through a "klein- moebius link" thus forming a more complex Klein  bottle structure; and would the simplest of these be a Lorenz surface? (A lorenz surface being the spaciometric surface formed by a Lorenz solution filament in space).

Feynman remarked to his wife that 1/243 had an amusing decimal expansion

1/243 = 0.00411522633744...

And there are many other anecdotes to the man's intuition and insight, and inspiration.

He inspired me from just listening to a few youtubes by him. Any way i found my number  and it is 2π-6.

I am going to call it a "spiral number" in the hopes it may turn up again when i do sort out a spiral reference frame to impose on the ratio tensor sructure from the Logos Response.

2*pi-6=0.2831853071795864769252867665590057683944

1/(2*pi-6)=3.531256652965522884896502576285279021366

10*(2*pi-6)-e*cos(√29)-e*sin(√7)=0.00009123888935780705434713174019675644282 (using degree angle measure)

and from i can derive this linking rule =

And from this i can mimic = where N is an integer, and F(x,y) is a general rule linking x and y parameters to a value parameter.

Why do mathematicians always use the same notation for 2 distinct things though related? this is what makes the brain ache! F(x,y) is the value parameters rule, the value parameter can be called anything, but its function(role) is to provide values for the working of the linking rule. It is these steps within steps that are so clearly iterative and linked that need to be celebrated as iteration, and connected to dance and rhythm and movement and music an colour if the bauty of them is to be appreciated, not balked at.

the rule is so sturdy that different linking rules can be put in each of its places and different integers. My favourite at the moment is sinx and siny.

z=x*y/(1-3/(3+x*y))

z=sin(x)/(1-3/(3+sin(y)))

z=e^x/(1-2/(2+e^x))

z=x/(1-2/(2+x))

z=x/(1-2/(3+x))

z=sin(1+1*pi)*y+cos(3+0*pi+1*pi^2)*x*y

This s trying out the aggregation based on mod(π^n). which has a natural home inside the brackets of the trig functions. This particular rule procduces a saddle rhombus, and this one produces a rhombus which when clipped is a perfect hexagon:

z=sin(1+1*pi)*y+cos(3+0*pi+1*pi^2)*x

So i guess that 2π-6 is a fantastic spiral number from the results i am getting when parametrising, using it. Its application is broad enough to include knots and weaves as well as spirals. So i am wondering if this number has anything to do with winding numbers in string theory.

From the relationships above it is no suprise that e is also a fantastic spiral number, but more used for knots and cones, than windings.

(http://www.fractalforums.com/gallery/3/410_08_08_10_9_57_57_4.png).

Graphed using free version of excellent Runiter 3d grapher.

There are a lot of things of interest in the current search for a grand unifying theory, and many theorists have conjectured from before Maxwell to now. The formulation of Coulomb's Law and Newton's law is so striking as to be deliberate. That Newton attempted to relate point masses and Coulomb point charges does not excuse the obvious abstraction in the formulae.

The formulae "speak" the same language, the linking rule for the parametric ingredients is the same, but the overall recipe is different, and one gives a gravity plum(b) cake and the other an electrostatic seed cake. :embarrass:

Nevertheless i have found some who unify (http://www.gravityunified.com/) the two and some who do not. (http://docs.google.com/viewer?a=v&q=cache:1r3TSJX6eIQJ:www.math.utah.edu/~gold/doc/grav.pdf+coulomb+gravity&hl=en&pid=bl&srcid=ADGEESgzxeurSmPwHTy-9RiHseFX7Vdn88p20vLW7qLTXMxidvKlUXCsz0t5JlvOHjb9xHOUDDaFtlg_EUImVqFEKAdb7ggFN2Bg1HqgHmYq2wkOl1R5S4Co_-vEkfGczA7FJpnjS4rb&sig=AHIEtbTahp7r-kLwenq_mLVFnd18rxCV0w)

I suspect that the difference is in the ingredients that are left out like dimension, spin, charge distribution, permittivty (density and viscosity), and vorticular Coriolis effects, and the direction of gravity's operation.The bond energy Quanta for nuclear and chemical interaction and stability also need there place in the formalism.

We have no other explanation for attraction other than force fields, but the form of these force fields is what i being investigated. My speculation is that the vortex torus form will be found to be the best fit and the best explanation of attraction and repulsion. Gravitational attraction will be seen as a spaciometric equilibrium of the two in a distributed regional field skewed by elemental and compound chemical reactions: the electromagnetohydrodynamic laws of order, which eventually  leads to a quantum value that repulses the bound mass according to the Coulomb law, which is generally an explosive decomposition of the regional order.

It looks to me like the "Sumerians" and the Harappians began the forms that would later become the subjec of polynomials, by devising tables of squares and cubes and even higher powers. And the Harappians were familir with the sphere the cube and the tetrahedron/cone.

All were intimately familiar with the spiral, but mythologically the magi appeared to hanker after the astronomical forms of the circle or ring while some Dravidian cultures revered the spiral or serpent, a more earthly form. It is only as one travels further eastward towards Burma and China that the spiral/snake is given a truly cosmological significance, and finally combined in the Yin Yang philosophy of chinese magi/sages.

The ring/circle as a magi-cal protection stems from this style of reasoning/analysis and heavily influenced western culture, whereas the spiral/snake concept informed eastern culture.

The fractal nature of polynomial bundling appears to have been appreciated more in southern regional cultures, particularly african where arrangements of campsites can show a self similarity in the design that is not only artistic but geopolitical: the status within the community is indicated by the position in the fractal and the complexity of the fractal design of ones dwelling.

This trait can be seen in all communities plant or animal, but the distinction between human animals is whether the spiral or circle is the fundamental paradigm. Those that adopt the circle adopt perfect abstract forms in their designs, those that adopt the spiral have more natural curvilinear forms.

The proto-polynomial forms of the Sumerians / Dravidians, bases around their base 60 (mod 60^n) aggregate system tended to represent regular arrangements of squares and cubes in a fractal array, with sectors of the circle typifying the perfect fractal arrangement into divisions of 60 (secs of arc, minutes of arc degrees of arc). These special polynomials were used to base their number naming system on, and to aggregate their value namespace, and to order their whole value and rank systems. By arranging their aggregates in this way they formed the first place value systems and the first power series systems of aggregation which are truly polynomial in design and use.

Thus their counting iteration was entirely consistent with the myths they had of the cosmos and truely magi-ical, but because it did not take account of the fundamental "spiral" form/ the foundational vortex-torus form as i now percieve it , their cosmological sytem always had to be corrected as it "spiraled" ( we now call it precession) out of sync.

I have tended to ignore anticyclonic vortices, under the mistaken idea that they were not common and shortlived. However though not as stable as the cyclonic vortices in our atmosphere they are common and reasonably long lived. In fact the red spot on jupiter is reputed to be an anticyclonic vortex.

When i was flying my kite it dawned on me that the air around me did not blow as in a jetstream but in eddies and vortices, somr vertical some horizontal and some at all "angles". And these eddies transport air from low to high pressure to low pressure eddies by swings and roundabouts. The motion is always spiral or coriolis and the strength of the wind gusts depends on how big the eddies are, what the density radiation rate is, the viscosity and elasticity of the air and the temperature and pressure differentials between regions at all levels above sea level.

Ed Lorenz in modeling these systems single handedly contributed to the  modern notion of chaos and non periodicity, but also established the overarching role of the vortex in making sense of all these motions and relationships.

Anticyclonic vortices spiral outwards, but i have more to learn about how they relate to cyclonic vortices and whether they too have the torus helix form.

Spiralman (http://www.spiralodyssey.com/PAGE11.html) has constructed spiral reference frameworks based on orthogonal spirals which are damn nice . I will explore these in time, but i am slowly adjusting to the fact that the spiral is not in competition with the circle. It wins hands down anyway, but the circle is a special spiral form, even one of the perfect forms that some look for.

In trying to standardise the building blocks of a spiral tensor reference framework it slowly dawns that spiral forms cannot be standardised like circles , cubes etc because a spiral has no constant defining unit! everything varies in a spiral reference framework!

I understand why cultural norms have been based on these standard units that do not vary, but this has denied us the tools to reference the experiential continuum easily.

From spaciometry i know that extension and rotation are the fundamental motions, and to reference spaciometric rotation a form is needed, which at the barest minimum is to standard rod extensions,joined at one end. The rods are thus relative and free to move the other end in any motion. If i relax the rod to an elastic material then i introduce a chaotic order of another rank!

So it is clear that even at this bare minimum  conditions of dynamic / mechanical consistency are to be taken into account.

However it became a point of interest as to how the cartesian tensor was formed, and the ubiquity of the triangle becomes apparent, but nowhere to be seen. One has to survey crystal lattices to find anything close to a triangle in nature .

 Clearly the discovery of the right angle and from there the triangle and from that the right angled triangle, lead over time to the general notion of an angle and then eventually to a notion of rotation around a circle. Angles and triangles appear naturally only in shadow casting when measuring the astronomy of the sun , and the massive assumptions in deriving angle and triangle from these practices pervade all our thinking about the environment i live in.

In a shadowcasting exploration the stick remains constant the shadow of the stick varies, almost elastically! But we agree that the ratios are in proportion, so the variation is not chaotic.

Thus spaciometrically the two rods is not even a natural occurrence! One rod and one piece of elastic seems to be the basis of all reference framework tensors! The triangle form is the only tool that keeps the elastic under strict proportional control!

With that in mind it would appear that the triangle and the spiral are two necessary forms for a spiral reference framework. This , because of the angle measure, implies a special spiral form called the circle is also required and of course one side of the triangle will need to be elastic.

Can i really reference all things by this set of tools and constraints?

Archimedes spiral (http://www.bookrags.com/research/archimedes-spiral-wom/). I like his thinking!


Although not at all rigorous  this happens when one converts a circle to a supposed right angled triangle with a height = radius and a base= circumference

Area = 1/2*c*r=π*r^2.

Geometrically the deformation to transform a circle into a right triangle is fascinating requiring the centre to unravel into a line in some way that is not obvious. No wonder Archimedes saw potential in the spiral!

Surds (http://docs.google.com/viewer?a=v&q=cache:EZ3FW2vvj2oJ:extranet.edfac.unimelb.edu.au/DSME/RITEMATHS/general_access/curriculum_resources/surd_delights/2005MAV_stacey_price.pdf+triangle+and+spiral&hl=en&pid=bl&srcid=ADGEESgCUPKCnDiNskFiaD3J5rmS0K_4yxQVd6QWKYxmrh8a4hDHe9NLee5-HEVsSUimeiAqmZwDcjrAr2g9Arp1EUpifMHDknY6UgwgJpsGPFtgPkmb-JN9hS3eHk2XHmvLyO1JDrH7&sig=AHIEtbQIp2aaTvT2PXTTC3rN2Dg1zUOl5g)

geometry (http://www.borderschess.org/circles-squares_spiral.htm)

primes (http://www.fortunecity.com/emachines/e11/86/padovan.html) and more prime spirals (http://buckydome.com/math/ulam/triangle.htm)

spiral numbers (http://www.mathpages.com/home/kmath620/kmath620.htm) and crop circles (http://temporarytemples.blogware.com/blog/_archives/2008/6/10/3737204.html)

tensor structures for data (http://docs.google.com/viewer?a=v&q=cache:DuqgbrZQyowJ:mapcontext.com/autocarto/proceedings/auto-carto-8/pdf/the-inward-spiral-method-an-improved-tin-generation-technique-and-data-structure-for-land-planning-applications.pdf+triangle+and+spiral&hl=en&pid=bl&srcid=ADGEESgcX4P3LjYjJS3pb7yGKQunYHxFTXrNMpvG-V2Py80xbqBazt5GuiWM4Cy8jmFmqGsg0R3_JmmL0pr-KAofWxxqhf2txuFQgwFNPA8589TjJcS8XEFmxh6KPpJ0IpEti40lStAv&sig=AHIEtbS42x9WBKCrF6jpuHLCrVNtLoTjDQ)


Remember......
  Remember what?

         Remember this:
               "our lives do not spiral out of control! (http://curvebank.calstatela.edu/baravelle/baravelle.htm)  click to find out
           " Rather, they spiral into  
   "an exhilirating rhythm of existence
"which is uniquely  
 "our  
0wn
"
'

http://upload.wikimedia.org/wikipedia/commons/9/9f/Spiral_of_Theodorus.svg (http://upload.wikimedia.org/wikipedia/commons/9/9f/Spiral_of_Theodorus.svg)
(http://www.fractalforums.com/gallery/3/410_25_08_10_7_53_17_0.png)
Not much survives of Theodorus' work but this is his legacy. My speculation is that Theodorus advanced the greek conception of π.

I think it was Theodorus who turned the quadrature of the circle into the right angled triangle transformation of the circle, and he did it to understand these ratios 1:√2.

These are called surds, measurements that have no archimedian proportions as they are now called. These values cannot be ratioed by finite archimedian values. However as a student of Theodorus Archimeddes appreciated the spiral form of his teachers "proof" that these values are measurable but not rational in the common usage. It is ascribed to Archimedes that quantities must be "mensurable", that is not infinitely large or infinitely small, but i think he drew on the work of Theodorus.

Theodorus and many greek mathematicians were enamoured of the triangle and its ability to decompose other forms into these basic forms. Thus it did not take long for Theodorus to realise that he could decompose the circle into a right angled triangle of the same area. If you roll the disc a full turn on its circumference you can see each bit of the area of a circle enter into a rectangle of height the radius and length the circumference.

Imagine as a the disc starts with 1/4 of the area of the circle in the box; a 1/4 turn brings in 1/2 the circle area, a 1/2 turn brings the area up to 3/4 and a 3/4 turn brings in the whole area and the final turn brings in another 1/4 area. This manoeuvre actually covers twice the area of a circle as Theodorus realised, thus the area of a circle was  1/2*r*c

What was and is more complex is to demonstrate this geometrically by triangle decomposition, this involved the ratios 1:√2 and others like 1/√2:1, hence his "surd calculator".

Archimedes immediately appreciates the basket weavers craft and the area of a circle and hoped to use the decomposition into a spiral to solve the problem.

This unfortunately was more awkward than he anticipated and so he never pursued it more, but his fascination with the spiral is well attested.

It is possible using Theodorus surd calculator to approximate the area of a unit  circle using √39 or√40.

√39+(√40-√39)/2-2pi=0.0015913521879919579970533383436099958616 =√40-(√40-√39)/2-2pi (pretty damn close!)

i have to work out how to post some spiral numbers i have done on spacetime.us as the app is awesome. The spiral numbers and the yin yang popped up unexpectedly in one this morning.

I have also just realised that a tailor's tape would be ideal for measuring lengths along a spiral reference frame. Spiralmans frameworks are where i will begin to explore,but already only two orthogonal spirals are needed to reference every point in space. The third orthogonal spiral i would use to measure orientation of the other two. thus a reference framework  is entirely possible where the parameters are displacements along orthogonal spirals. Spiralman (http://www.spiralodyssey.com/PAGE9.html) has frameworks for the Archimedian, logarithmic, and golden ratio, fermat spiral which i will denote by .

The ear (http://www.daviddarling.info/encyclopedia/I/inner_ear.html).

(http://www.fractalforums.com/gallery/3/410_26_08_10_3_35_25_0.png)

The Logos Response is based on the activity and functioning of the sensors, has a major contribution from the ear, and the structures in it are really interesting.

The two main structures that stand out are the semi circular canals and the cochlea. These arise from 2 "chambers" the uticle and the saccule. These join to the one auditory nerve conduit, but clearly two bundles of nerve fibres use this conduit. There probably is preprocesing that occurs along the length of this conduit before they arrive at the main brain centres for processing.

So we are already considering a ratio between the uticle and the saccule inputs from our auditory system.

The two spatial and special forms you will notice are spirals. The cochlea is a spiral acoustic chamber which is so neat! it is breath taking to think that a sea shell informs us of how our own cochlea works. This is crazy, because in one form we have a "speaker"/ sampling system that analogically seperates the pitches and amplitudes of sound to give us that crystal clear quality of pitch ratio distinction: we can hear the high notes and the low notes and those of "Mr in between". The note blends and chords and note boundaries are all distinguishable by this system. What a great natural design for a speaker system!

The spiral form orders these possibly chaotic harmonies into a ratioed musical system. A damaged cochlea, then, may mean that harmonic sound distinctions are confusing or non existent. WE normally have 2 of these. Interference phenomena among ratios seems to be a crucial processing strategy for apprehending the set notFS in a richer more informative way, and certainly stereophonic, sonar detection would not be possible without it.

That brings us on to the next spiral forms the logarithmic loop canals (so called semi circular) which are arranged orthogonally from the uticle or nearly orthogonally. How crazy is that! That is so neat and mind blowing, to think that orthogonality or near orthogonality in reality is the basis of our spatial awareness and sense of rotation and orientation in this fundamental way.

One cannot hlp but be struck at how our biological functioning imposes itself on our mathematical constructions,tools vis a vis the Cartesian tensor, the polar coordinate spherical coordinate tensors. This is to be expected as the logos response deals in comparison and distinction in ratios, thus the inate form of the ratios will always come through, as ratios like similar triangles for example are applied at all scales infinitely.

This does mean that our notions of notFS are exactly dependent on the ratios in our sensory systems, no necessarily on the ratios in notFS.  

So these logarithmic loops as orientation and rotation measures remind me of the spiral frameworks Spiralman (http://www.spiralodyssey.com/PAGE23.html) has been able to construct, and their function in the CNS will inform how i explore these frameworks and their use in relativistic motion.

Again we have 2 of these sensors and it makes me think of 2 gyroscopes and the interference pattern from their ratios what are they detecting?

One use suggests itself in the hunt for gravity waves, That not only should the current methods use laser coherence to detect stretching and shrinking of a long pipe under possible gravity wave influence but also the interference of gyroscopes along those lengths. As i understand it lasers and gyroscopes (http://en.wikipedia.org/wiki/Ring_laser_gyroscope) are being amalgamated more and more in research institutions.

The Logos Response compares and distinguishes Ratios, but the ratios are all related to one another, so focus is necessary to concentrate the attention on a particular region. But this means that the region focused on is in a ratio with the whole ratio data set from the logos response. Thus the ratios i may divine from a region of attention are in fact ratios within a ratio itself.

The Golden ratio (http://www.spiralodyssey.com/PAGE20.html) is perhaps a formal recognition of this fractal state of affairs.

(http://www.fractalforums.com/gallery/3/410_26_08_10_6_31_16_0.png)
The Auricle (http://www.daviddarling.info/encyclopedia/E/ear.html) has other structures which function in the logos response for vertebrates; resonance chambers, pressure amplifiers through lever actions,pressure equalisers and head tilt sensors,gyroscopic motion sensors, amplitude and frequency sensors.

Although i began my exploration of the sensory system with the eyes and discovered the logos response in that system, it i clear to me that the auricle system is fundamental in a way the visual is not. The sense of rotation lies in the auricle system and the sense of orientation lies in this system also at least the sense of head orientation vis a vis( or rather ratioed against) the proprioceptive sense of orientation. The auricle and kinesthetic senses of orientation are more fundamental than the visual reference of that orientation.

Thus in the auricle and propriceptive systems i find the fundamental notions of orientation and rotational movement relative to the organisms form and structure. If the visual sensors are combined i find that rotational movement and orientation can be referenced against a visual map which includes a map of the organisms form and structure.

Thus i note the systems fundamentally provide information /ratios about orientation and rotational movement  Where then does Extensional movement reside and the notion of extensional direction (from which and upon which we base the notion of Axis) ?

This is of interest to me because despite seeming fundamental Axis is not at all represented in the sensory system, and therefore seems to arise from the processing algorithms that form the basis of extension.

As far as i can tell the notion of extension arises because of the interference patterns produced by the binaural, stereoscopic, dual gustatory and rich proprioceptive web-maps of the inter-reacting sensory systems. The point here is that extension is a computational output that is at a different computational level to rotation and orientation. Thus extension will have more computational artefacts than orientation and rotational movement.

What this all means is that "Perspective" (as a relevant example is a computational effect) and the vanishing point are a computational effect. Of course the structural form of the eye contributes to this, and the ratio of "interference  pattern source" signal processing also contributes. Thus if i focus attention on the interference pattern at a particular region in the visual data/ratio streams the perspective and the detail and even the image size output changes to reflect that. The output to memory thus can significantly differ from the raw signal output from the sensors themselves.   However it is unlikely that the raw signal output will be interpretable without the processing that occurs to "make sense" of it so to put my point another way: the data that is currently being processed in your processor to produce this screen you are currently looking at will be unintelligible without the visual data processing algorithms that re-translate them to a visual image.
The perspectives in this visual image are not reflected in the raw data that is being processed as it may take only a few bits of machine code to describe the colour and area of the screen but masses of data points to modulate each pixel to give the correct amplitude and relative colour and persistence, etc. And of that vast ocean of data only the selected part is output to screen, and only in the window assigned to it, at the resolution assigned to it . Thus these very words are the focus of your attention, but the processor and arrays have present on the whole screen more information than you are actually focused on right now. Your perspective of he screen has thus been altered, so that your experience highlights these words and their references and not the physical display.

Their is no vanishing point on the physical display so every selected  thing that is currently being processed is flat in front of you, but by focusing you have made various parts of the screen vanish from attention ( this is now called attentional blindness). But what if the algorithms put a vanishing point on screen to reflect the computational ratio of the data being processed? Thus the processes with smaller cpu clock cycles  would be represented on the screen with smaller windows, and the larger cpu clock cycles with larger windows. The resulting quilt map would mono-scopically reference a ratio field. If these windows were arranged on a spiral we would observe a natural spiral vanishing point that would immediately make visual sense to us.

The point then is that the notion of extension is as much a computational effect as a "real" spatial arrangement in notFS, and the notion of Axis derives from this computational output. The cone effect of the vanishing point is likely to be a computational artefact of the arrangement of processing loads on the CNS distributive parallel computational system.

Thus despite the seeming fundamental nature of axis it is not as important as relativity, and the relational arrangement of certain structures within the organism are sufficient for rotational movement and orientation to make sense.

Now orientation as a spaciometric attribute relies, as it must, on an individual organism defining it and demonstrating it. It is a real thing only in this sense, and makes no sense as a line on a piece of paper.

However, if the defining organism can spaciometrically reference these marks and lines  they may have a significance as an orientation or set of orientations to another organism. You may have heard of the dance of the bumble bee? This is such a non human example of the general system i refer to, and exemplifies the basis underlying our graphical representation of orientation. When extension data/ ratios are combined with that orientation definition we have the basis for the notion of Axis.

It is worth noting that in none of this has it been necessary to define a surface called a plane. A plane therefore is even more of a computational artefact than an axisIf i spaciometrically rotate and keep the relative ratios within my organism the same i might visually reference a surface which i can define as a plane, but equally i might reference a surface which is a cone, or edges or surfaces of any form that encloses me. What i will not reference obviously is the spiral or vorticular nature of all my observations, in all sensory systems.

(http://www.fractalforums.com/gallery/3/1307_24_08_10_5_51_08.jpeg)

This is a 2d notion of a 3d DNA wrap.

Hi jeovajah,

I don't mean to disturb you and intervene in your blog (which is a little bit too complex for me I must admit). Let me know if it's OK or I will remove this post. I'm not a mathematician, and not a biologist, but I just wanted to share some thoughts here, in case you find them interesting.

When I see your comment about the DNA I can't help trying to find similarities between fractals and life. First I find it completely amazing that DNA is the unique genetic coding system for ALL species on Earth. Has it won the competition several billions years ago against other codes? If not, would it be the same code if we eventually find life on another planet? I think these are questions no one can answer (yet?)

What is even more amazing with DNA is that it was the first molecule complex enough to have this incredible power of self-replication (due to a very simple chemical concept : the fact that the ATCG bases work in pair). That's what we call "life", but it's just a step in the chemical complexity of our universe, it's not magical nor supernatural.

And I think we could draw a parallel between this concept of self-replication and the notion of iterations, so of fractals. Each time a cell splits in 2, it's just like an iteration. So all living creature are built thanks to an iteration process. So they're really fractal in essence, meaning that all the "program" to build them is included in the "seed", the DNA, and if you take any living cell (like if you zoom in a fractal), you could potentially extract its DNA and rebuild the full creature with it (like you can see minibrots everywhere in the Mandelbrot set).

Iterative process, self-similarity, "seed containing the program" (=fractal formula) : are there any other conditions to verify in order to validate that life is fractal ??

My 2 cents...

cheers
bib

Hey Bib , Welcome! :D

I thought some evil genius had stuck a sign on my Thread saying "mathematicians only" ! No way man! It is a thread after all and not a blog  :stickingouttongue:

So i welcome everyone :howdy:.

First let me apologise for my complexity. Although i accept the criticism/comment  i cannot promise to change much as the thoughts come to me as is, literally. I am more interested in capturing the thought than in making it intelligible to the reader, including myself! :whistle: :crazyeyes:

Anyway this is fun for me and a reference notepad of insights. Anybody can contribute , challenge correct, question (what happened to the alliteration?) , but i hope someone will want to collaborate. I do not have the answers and i do not intend to judge others contributions. You know the pedagogues of my day called teachers of math did that to so many that they have become timid and bereft of their mathematical heritage. Its a power trip man. Pure and simple! :order: And while i am on a roll i have a beef with Gauss for his treatment of Riemann! :devil:
 :rotating: Anyway now i got that off my chest i feel a lot better! :rotfl:

I think the more i learn about Feynman and Dirac the more i see their Autism, but Feynman was able to use his Aspberger"s in a field where it was valued and he could mature in a positive framework. Dirac had to overcome much more difficult circumstances and a different level of Autism. So now what about Stephen Hawkings?

Mathematical physicists from Newton to Feynman are my favourite maths teachers, but of course they can be self righteous too, a trait i think to be avoided.

So welcome, Welcome Welcome ! All are Welcome.

I want to post a note about the iterative structure of the rod and cone sensors and how they are formed for their function and provide stepwise variation in signal output which is akin to digital sampling. Thus the notion of analogue and digital evaporates as a biological sensor can be shown to demonstrate digital sampling like all of the camera sensors which are in use nowadays. Plus how a drop of oil provides an interferometer /diffraction system that isolates "colour" frequencies and enables a colour signal output from a cone, while a rod provides a contrast ratio  signal output.

It's all iteration, its all biological and it's all there, man!

The rod cell (http://www.daviddarling.info/encyclopedia/R/rod_cell.html)
(http://www.fractalforums.com/gallery/3/410_26_08_10_3_36_15_0.png)  .The cone cell (http://www.daviddarling.info/encyclopedia/C/cone_cell.html)

Image from The Internet Encyclopedia of Science (http://www.daviddarling.info/encyclopedia/R/retina.html).

The rod and the cone cell develop  the rod and the cone from a  cilium. This hair like structure is associated with the mitochondrial cell elements and are usually spiral or helical in form and function. Thus a cylindrical helix and a conical helix form are the precise descriptions of these cell segments.

The helical form enwraps regions which are here called flat and parallel unit membranes but which in fact are helical "slices". These slices contain the distribution of the photosensitive pigment; in the cone they contain 3 pigment types and in the helix just one.

This spiral arrangement of pigment provides the analogue to digital sampler framework. The heliical forms both the cone and the helix act as collectors of light signals (photons) and the light enters into them in the human eye from the base of the image. Immediately the role of the oil drop in the cone becomes apparrent as does the cone shape. The oil diffracts the light into its constituent colours and refracts it to different  parts of the cone. Thus each part of the cone registers different colours and has a different photosensitive pigment concentration. It also digitises the analogue signal to that part of the cone.

So the helical sensor due to its size and shape is capable of measuring intensity of light to a very fine contrast ratio and the analogue signal is digitised by the amount of pigment that gets changed increasing the electron transfer to the helical wrap spaciometrically. Thus the duration for a helical element to transmit its charge to the mitochondrial elements in the rod cell is also measured via the "spiral length".

Intense lights therefore convert more pigment and take longer to fade the signal output. The mitochondrial capacitors are slower at discharging because there is more of them, but the ones in the conical helix structure discharge more quickly because there is less of them. Consequently the conical system is faster and able to deal with finer detail as well as colour but not as sensitive to intensity or contrast as the helical structure because the "raw light" is impinging on the helical unit layers while the diffracted and thereby diminished light is impinging on only a part of a conical unit layer.

The complex functioning that produces The Logos Response i have discussed elsewhere, but suffice it to say that despite the sophistication of the sensor it is down to the processing and wiring as to how the signal is managed; and at every stage we can identify a reduction in informaton content as the signal is transduced to make its way to the CNS.

Thus whether we liike it or not the setFS is only a poor model of the set notFS, and human beings who claim extra perception may in fact be able to demonstrate this in their wiring diagrams, processing algorithms and perception schemas, to name a few.

My joy however is to point out the presence of spiral and vorticular structures in the biological solution to signal processing and to suggest that these forms may be useful in modern electronics.

"Imagine as a/ the disc starts with 1/4 of the area of the circle in the box; a 1/4 turn brings in 1/2 the circle area, a 1/2 turn brings the area up to 3/4 and a 3/4 turn brings in the whole area and the final turn brings in another 1/4 area. This manoeuvre actually covers twice the area of a circle as Theodorus realised, thus the area of a circle was  1/2*r*c"

This maneuver is not unique to the circle, but in fact relates the perimeter of a convex 2d shape to its area. By finding the geometric centre of a shape and measuring the perpendicular distance to one of it sides that gives the height of the rectangle measure the shape will rotate into along its perimeter. The area of the rectangle is twice the area of the shape.

Hallo Jehovajah,

thanks for all the commends on my work here in this forum.
When I read this thread I get a lot of new Idears cause some things are very fundamental but I have a lot of Problems to followe your train of thoughts.

So let me ask you at the begining some simple Questions:
What does the abreviation FS mean? Fractal System?

For me a set is something like this:
{1, 2, 3, 4, 5, ...}
But what is a universal set?

For me a fractal system is something like:
z_n+1 = z_n² + c;

When I look on the set {1, 2, 3, 4, 5, ...}
I have the Problem with the ... which means Infinit, which is something I canot measure and what is not computable.
So I would prefere to have something like:
{1, 2, 3, 4, 5, ... n}
From this kind of set it is easy to come to the set:
{1, 2, 3, 4, 4, ... n+1}
So I will be possible to define a lot of Maths in a recursive way like a fractal.

Is this the direction in which your idears go?

Hello Hermann

I apologise frequently to you and others about the difficulty in following my thread, or particularly my train of thought. In part as i have explained before this is due to the intuitive nature of the writing, and the declarative style that i seem to have. I have also to point out that for me this is a journal, a noting of random thoughts and intuitions that drive me to write before the notion fades back into the vast computation that surrounds me, and of which i partake but a few morsels, crumbs of consolation for my tardiness and human frailties!

Also, as for a long time no one has been moved to interact with the thread i have not needed to clarify anything much except when i am moved to do further work on an idea. Sometimes i find myself inspired to write a full description of an idea, a truly satisfying or sometimes driven experience, only to find i have written the idea down somewhere else already! Some of the ideas i admit would be better spun off into their own threads, but i do not propose to impose on our host by multiplying topics endlessly.

Because this thread purports to be about Axioms i have had to spin off the axioms just to provide a more straightforward access to them, but as you will notice my housekeeping is not the best, and i need to tidy up some of the numbering system. I will get to it, just as i eventually get to my typos, which are frequent and occasionally interesting: for example Axiom!  :embarrass: i have never changed it because it says something i cannot define! ;D

So when i started it was not out of the blue but after years of self reflection and working through practical philosophical issues about the nature of knowing: Epistemology and religion and language and communication etc.

Man i had a lot of issues! But any way the epistemology had to start with me, because i had no idea if anything else existed independently of me. But of course this is not satisfactory because "me" is a construct developed over "time", etc and "space" is a construct i develop over time.

I had got as far as the concepts of infinite possibility space condensing into probability space instancing in a particular statistical reality space,from which and in which i have my experiential continuum. This was driven by a common religious concept of "ALL in ALL", and "in whom we have our being". Also the infinite pattern formations were inescapable, but the epistemology of pattern recognition or perception was not clear.

Then i came upon Fractals, and immediately found a resonance. Space then became infinite fractal possibility space, etc.. I literally woke as if from a dream and began to look at this thing called fractal. Who should pop up bur Benoit Mandelbrot and then Arthue C Clarke. Arthue i could understand having read his science fiction from a child. Benoit i had come across earlier in life, seen the clouds and the mountains , been put off by the mathematics and put it back in the library as crazy abstract mathematicians stuff! (it seemed arch and Facile and it still does, but at least i have the resources now to get at the basis of it- i intend to explore dimensionality along with parametrisation).

I had a long term beef about numbers and why "i" could not possibly be a number imaginary or otherwise. Thus i started my exploration into FractalSpace, and the foundations of maths, seeing that "Fractals" were fundamental to the order of all things.

I start off all Axiom this and corollary that, quite the mathematician :-*, darling! A bit up myself?  Not really, just Autistic and not sure where i was going who i would meet along the way, how out of date my ruminations may be etc. I had assumed that the artists here would all be jobbing mathematicians. How ironic and special to find that they were players and playful! Just what i needed.

The thoughts range over my areas of insight and interest, and some do not apparently appear mathematical. They probably "ain't" in the old sense, but since i discovered, uncovered The Logos Response all things are mathematical or strictly "ratios".

My current aims are to fully explore the Logos Response and its impact on Epistemology, to continue to review the basic "mathematical" structures of which Spaciometry is primary as a fundamental output of the Logos Response, and in conjunction with spaciometry algebraic thinking. Algebraic thinking is how i formally construct and apply and deploy axioms, definitions, transformations, products of transformations, theorems, notation and rules of computation or manipulation,mapping, iteration, boundarisation, enumeration and transformation rules.(Some of these overlap).

 from these Algebras i derive specific arithmetics which are the fundamental computational tools i use in everyday calculations of one sort or another. Arithmetics can be very intense and algebraic at times, so i welcome the tools of computation now available, especially symbolic logic applications.

Calculus is usually marked out on its own but is a type of arithmetic for dealing with change, so modern computing platforms that can animate those changes are a fantastic aid.

I have found out a few things of great interest so far, and i need to note something in more detail about the change from ratios to Fractions, and thus the whole misleading conception of number leading to the complex numbers so called.

A lot of the "genius" of early mathematicians is due to the fact that they thought in and were fluent with ratios and proportions. For example i recently found that the greeks were not squaring the circle to find its area they were triangulising it (if such a word exist!). This lead me to observe a curious fact between rotating the perimeter of a convex 2d shape into a rectangular area, and the area of the shape. I am currently wondering if a similar relationship exists between surface area and volume, or is it just rotation around a given axis.

Hope you find this helpful Hermann. Essentially the whole of my thoughts are to explore as rigorously as i can the fundamental function of iteration in defining and developing our mathematical ideas, and as a consequence the inevitable fractal nature of everything, but this time not in Benoit"s sense but the modern sense of self referencing, self similar patterns at all scales. :dink:

Not really. Set theoretic descriptions are a language algebra with operators in the notation and rules of the set. The set you denote    (see how the language has to change to become rigorous?) is not a real set except by definition. You won't find it at the bottom of your garden for example. You won't even find it among communities of religious monks dedicated to chanting and counting! So what is it you are denoting? My contention is that this is not a set but a cultural iteration called by me +1. But now perception comes into it. What do you perceive you are looking at? I do not know, and any way is it the same as what i perceive or intended to communicate? Again i do not know.

Only yesterday i was struck by the enormous assumption we are taught to make through number bonds: 20 *20 = 400. Is it really? What the hell have i just written? some marks on your screen, some buttons i have just pushed, and all of us reading that agree that it is correct. How did i do that?  Did you do in your head what i did? i did 2*2 =4 put 00 on the end you get 400, but i could have done 2+2=4 ,10*10=100, these are 4 hundreds, that gives 400! i might even have done 2+2=4 put 00 on the end that gives 400. Why do these numbers do that? Why is one wrong and the other right, but they both give me the right answer?

So i never could understand Russels Problem, until i realised it stemmed from the flip flop of undecidability. Undecidability exists in the real world and we solve it by making a real decision. The real decision is a bit like Archimedes principle, we avoid these "unreal" ,not everyday abstractions and deal with what is quantifiable. This is what you have done. But hey when you make that decision you set up consequences that shape your world, a bit like Schroedinger's cat!

The only set of all things that can include itself is the set of all things. And as soon as you realise that you realise the iterative nature of all things and how one condition drops out a universe of consequences from that set which partition that set of all things. Thus as soon as consciousness or perception is allowed the set of all things is partitioned by that perception. This essentially and abstractly is the effect of the Logos Response: the set of all things is ratioed, that is partitioned, proportioned and related irreducibly in some way to a portion of the whole.

I find now that i am comfortable with iteration being foundational to every perception and notation, and thus the set with the ellipsis means an iterative process is being evoked to describe what goes in between the brackets. Your alternative definition is equally valid and in fact i would define away the issue by formally making them equivalent, much in the same way that 0.99999999999999....... is formally equivalent to 1.

My whole thesis if you like is that mathematics is a recursive system, that is all of mathematics is iterative, and finite systems are only stages in the overall scheme of things.The setFS was initially going to be a Universal set, but then i realised that it was really a model of another set that is universal: notFS. SetFS then has really become my mental model of reality if you like, and the place where all paradoxes live! Thus it represents discovered knowledge particularly overtly iterative.

There is a lot more that can be done to make this obvious, and a lot of overlap with computability. I am happy and grown up enough to accept maths as a subfield of computing, despite the obvious irony. In "truth" maths has always been at the behest of one patron after another, like music,so where is the problem?

I can begin to address the issue of the concept number. At the heart of it must be spaciometry. I have abstractly refered to the tensor spaciometry as quantity and "number". The relational ratios in a tensor encapsulaing Quantity and the boundary of the tensor encpsulating unity or one.

Numbering is a simple naming activity, but the naming activity can be made complex or rhythmic or repetitious or systematic. It is these intuitions that cultures bring to their numbering that inform their concept of number. Value is also attributed to their cultural counting /naming iteration, and that value is proprtioned throughout the whole process of numbering so that each number may hold an ordinal value or a cardinal value or both, and a rank according to the proportion of value. The map between value and number is spaciometric and thus provides a circular or tautological basis to value. The source of value lies within our own neurology and is culturally maintained, defined and standardised and enforced, at least in weights and measures and SI units etc.

Thus number becomes a name that identifies a stage in a cultural iteration onto which a culture encrusts many meanings, all of which reflect a spaciometric attribute of the many tensors in space.

Given this description i venture to add that the attempt to tie number mathematically to one abstract tensor, a linear fractal called a line distorted the concept of number and confused those who had cultural attachments to number. Newtons tutor John Wallis (http://en.wikipedia.org/wiki/John_Wallis) was principal in achieving this and despite the neatness of it the underlying fractal has come to the fore when mathematicians were not ready for them in general. Thus Cantor, Julia, sierpinski, Peano, all gutturally felt these fractals as monsters and horrors eating away at the basis of reality and of course mathematics.

The area of solving arithmetic problems using algorithms led to the development of Babylonian binomial equations to trinomial and quartic and eventually quintic. The increase in the number of terms in the equation reflected the effect of iteration on these algorithms as they described relational aspects in spaciometry,and the systematic relations that underlie manipulations. This "attacking" of a problem by "manipulations" is a very militaristic paradigm, and underlies all the notions f combination and permutation, issues that would very much concern the militaristic mind through the ages, but also the commercial or merchant mind would consider these aspects of the spaciometric tensors under its hand.

This rich appreciation of number and value is what the number line threatened, and that is why it was a tool for mathematicians per se. Fractions and the numberline are where mathematicians withdrew contact with the general culture and began to distinguish mathematics as a specialist field of study with certain enforced tools.

Fortunately for us the iterative nature of reality put a cold hand of dread on them and hopefully will prevent mathematicians from disappering up heir own anus!

So in the times of the great Taxonomists (http://www.nhm.ac.uk/nature-online/science-of-natural-history/taxonomy-systematics/history-taxonomy/session1/index.html) the subject of mathematics came under taxanomic scrutiny, and among other things the taxonomy of equations was updated to "polynomials" (http://mathforum.org/library/drmath/view/69475.html). Mathematical reference for the body of knowledge to do with algorithmic solutions to quadratic, cubic, quartic and quintic equations became subsumed under the heading polynomial of rank or order 2,3,4,5 etc. The term binomial had existed prior to this for a while , so this represented a tidying up of the taxonomy for ontological purposes.

Early on in the development of the solution to the equations surds had been encountered as solutions. Surds are purely geometrical values, in that they naturally arise in euclidean geometry of the right angled triangle. The very name of the equations quadratic and cubic testify to the geometrical basis of these algorithms. Going beyond the cubic meant that no geometry informed the solution, Thus it made solution harder and less intuitive and relied much upon "abstract" relationships and symbolic manipulations,and analogy of form. Essentially try to view the quartic and quintic equation as some kind of quadratic or cubic one. That is simply to utilise the spaciometry of the day to intuit the solution.

Without formally recognising the difference mathematicians had come across a type of value in solving their equations which were geometrical, ratioed and measured, not counted. Thus they were not numbers, nor were they the ratio of any common or archimedian numbers. They were thus called surds and meant "geometrical measurements".

In the course of this feverish activity mathematicians (http://1iz.heimat.eu/history.html) came across a curious surd √-1. As mathematicians new general surds had a value they did not reject this as meaningless, but as some geometical measurement they did not yet understand. They were necessary for many solutions of quartic and cubic equations and so had an algorithmic value.

It was not until Argand that their geometric meaning was hinted at, and by then the number line had queered the pitch and the surds had become irrational Numbers, rather than geometrical measurements. it could not be seen for a long while that √-1 was a geometrical measurement of rotation. It is still not appreciated as that even today.

Due to abstract and symbolic manipulations some mathematicians (http://en.wikipedia.org/wiki/Negative_and_non-negative_numbers#History) had developed algorithms that gave solutions to the quadratic and cubics both as numbers and surds, particularly when the negative number rules had become well established. The negative numbers were another geometrical value, but because they were defined  in terms of a balance, and from that the commercial bookkeepers financial sheets/ tablets, their geometric meaning  was obscured. Their geometric meaning is in fact a rotation through π radians, if one accepts the number line.

The chinese and the indian mathematicians had a good understanding of them in the commercial context, and in the context of quadratic equations, but they were not easy to accept just as √-1 was not easy to accept.

It is of great importance to realise the spaciometric origin of these quantites and how they do not exist without the awareness of a mathematician and his/her paradigm. The concept of number is a cultural totem, based on identifiable tensors in space. The relational ratios in a tensor quantity are key to the distinction i am about to make:measurement and distinction .

The Logos Response provides me with measurements of ratios. These ratios are a field effect in my experiential continuum, and i respond to them by processes within my CNS and Peripheral NS with an action that boundarises regions in that field, based on comparison of the relativistic motion attributes within those regions. Thus the field of ratios from the Logos Response is a Motion Field. Although i cannot say much more about that yet i am working on it in the thread on the Axioms of setFS. Nevertheless the point is that Measurements of ratios not Counting is the fundamental response to the motion field in the set notFS.

The distinctions we make by boundarisation are the source of our language response. Thus our language response holds the bounded distinctions in and among  the tensors. One aspect of our language response is the identification of plurality, which means the recognition of more than one and the recognition of repetiton of bounded regions: identical, similar, or none similar. At the same time i recognise the relativistic relationships between these regions in this plurality. Thus the spatial arrangement is inherent within this notion of plurality. Thus to sum it up almost the first notion that arises through the logs response is a measurable spaciometry; the second notion is a languaged spaciometry and the third notion is a countable spaciometry, in that order.

All Founding mathematician exclusively engaged with the spaciometry in doing and thinking about their mathematics. Thus while a region is real when it is in front of one, it is also a real memory that can be in front of the mind at the same time , Recognising this the indian mathematician in particular were able to conceive of debt as an absent region, a re-balancing of scales or the filling in of a hole, or the removal or changing of a colour. The chinese used coloured rods to represent a region that was removed from the direct view of the mathematician, but was important to account for the regions in view. Spaciometrically the red rods were removed from the relationships under purview, but needed to be accounted for. Each red rod thus told a story, and the story might be one of debt, loss, investment, advance or retreat, whatever the mathematician wanted to account for over a sequence of events. a set of relativistic motions.

With these spaciometric tools and memory tools in mind Indian mathematicians were able to give rules of manipulation which became the rules we use for signs today. How they arrived at -*-=+ i have yet to uncover, but our mathematics is the way it is because of this rule. We now can explore different "sign" rules and see what mathematics they produce, but the one we have resonates in so many ways with the natural order that it is unlikely to be replaced.

Thus the geometric/ spaciometric underpinnings of number are clear but mathematicians began to confuse measurement and a number concept. The geometric measuring scalar fractal was and is different to the number line concept, although of course the number line concept is an analogous system that principally John Wallis used to great effect. However Wallis used it as a geometric measuring line, later mathematicians like Cauchy and Dedekind ripped it away from these geometrical roots and created the number concept we use today. This number concept is a aggregation of number, memory tools like number, surds and Fractions, and infinitesimal like limit values including continued fractions and e and π cognates.

It seemed crazy to extend numbers by fractions, even crazier by negative numbers, but to then attempt to add √-1 was a step too far. Mathematicians have resolved the conflict by the invention of vector mathematics, but few recognise the work of Bombelli (http://1iz.heimat.eu/history.html) 

1572 book-keeping was highly
developed in northern Italy,
but even "simple" negative numbers were just introduced
(and the + and - signs unknown). So the hydraulic ingenieur
Bombelli  wrote a poem about "piu" and "meno" to teach
calculating these.  In his book "L Algebra" he didn't try to
solve x²+1=0 any longer; instead he recognized the "necessarissimi"
existence of squareroots of negative numbers and introduced
sign-rules,f.e.:

(http://1iz.heimat.eu/i-page-Dateien/piusqrt.gif)
Just insert numbers and you can calculate every combination.
So he introduces them as new members to the family of numbers,
or, more precise, of quantities, lets say of a different branch (we
express this by the word adjungate or adjoin). In modern words
he is calculating vectors. For him, it was not an abstract construction.
Solving equations were done with geometric constructions and
Bombelli used L-shaped rulers for this:

The aftermath of this age long construction has been a confusion in the concept of number, carried on today through the use of number when referring to geometric values and operations, and relationships. The concept of a tensor has the power to resolve this issue of geometrical measurements in a vector type relationship, or a matrix or indeed a relational database called a tensor. This allows quantities and measurements to be separated from number and number to be returned to its cultural role in naming stages in the counting iteration.

Tensors, by which i mean weights and measures and dimensional units are the geometrical heirs of the "number line" concept, modified now to a geometrical vector.

Hello jehovajah,

thank you for the survey you have written for me.
But I thing it is bejond my time, energy, an knowledge of mathematics to understand your writing.

It is very inspirational reading your writing, even if I find more questions and answers.
Maybe I can elaborate some of my idears and post them here.

For sure, Hermann, 2 heads are better than one they say :D

Welcome aboard! :surf:

http://www.fractalforums.com/3d-fractal-generation/truerer-true-3d-mandelbrot-fractal-(search-for-the-holy-grail-continues)/60/

In the light of further research i appreciate that the rotational and extensional aspect of z^2+c is in fact codefying spiral orbits, as a rotation while extending is precisely a form of Spiral. Thus the mandelbrot set is what remains after vectors c  trace spiral paths through the vector field with a sculpting effect whenever |c|≥2.

This i think is what Vector (http://www.fractalforums.com/index.php?action=dlattach;topic=2035.0;attach=1310;image) is exploring, but the spiral orbits exist in z^n+c where n>1 regardless of changing power or logarithm.

Julia is a spiral orbit traced by the z vector, which is the stepped vector added to the same c vector every time after squaring, thus spiraling and translating.

When I read this thread a lot of idears come to my minds. So reading and understanding nothing makes me very creative.
To think about the fundations of mathematics a first question comes to my mind:

Are this to numbers equal?
1 = 1

This should be less a question than a starting point of an essay I have in mind and would like to write down.
I think I can not do this in a short time. But I can come back later and can change this post.
For this I have to write down my understanding of a matched filter in digital signal processing.
Then go to artifical neural systems and then one has to understand the behavior of real neural systems and our knowledge of the human brain.

Big a program. I will surly fail cause of the lack of time and energy.

But I can post some reverences to books I like:

The first one I used at University. It gave me a deep view in digital signal processing.
For me this book was a starting point to develop my idears in a scientific context.
Digital Filtering and Signal Processing
from Donald Childers and Allen Durling
the book is from 1975 an I think it is no longer available from the book shops.
It is more my personal begining of digital signal processing and not a starting point for learning digital signal processing in the year 2010.

A great inspiration for me was the reading of the Chapter 3 the design of digital filters.
I was very impressed by a recursive digital filter.
Where the output of the filter was again feed back to the input.
Building an infinit loop that can produce infinit patterns.
(May be equivalent to Stephen Wolframs rule 30 when setting the parameters right. (So I have discovered it first!))

The next book is:
Introduction to "Artificial Neural Systems" by
Jacek M.Zurada
This book lead my idears further in the direction of how information processing and system control can be done in biological systems.

The last book is:
The Priniples of Neural Science
from
Eric R.Kandel
James H.Schwartz
Thomas M.Jessell

I bought this book because I was very much impressed by the lecture that Prof. Kandel gave in the Iconic Turn lecture serial at the LMU in Munic.
=6&cHash=98c6b2bafc]http://lectures.iconic-turn.de/iconicturn/programm/video/?tx_aicommhbslectures_pi1[showUid]=6&cHash=98c6b2bafc (http://lectures.iconic-turn.de/iconicturn/programm/video/?tx_aicommhbslectures_pi1[showUid)

I have not read the book complete but I am always impressed how recursive processes are implemented in biological systems when only looking at the pictures.

So the geometrical /spaciometric concept of measure in the process of measurement is my foundation for aggregating "values" of various measures and forming a measure-line concept which includes the natural number names plus many more for example π, e. The point here is the measure line concept is a convenient organisation of spaciometric measures which uses numerals in a namespace to identify the measures specifically.

Remembering the fundamental process involves ratios was brought home to me by the following project.

I want o construct a protractor that measures in radians. A simple enough idea as most mathematicians measure in radians not degrees. To do so i came up with the notion of rolling a circle of circumference R around a circle of Radius R and marking off the cardioid points. Clearly this will lead to procession around the circle radius R, but i only want a simple radian measure at this stage.

The construction cannot be done by compass by the way, and the ratio is r : R = 1 : 2π. So i can use some euclidean properties to construct the measurement r given R. I need r to draw the circle to roll around the bigger one.

Firstly i choose an r and draw a circle. I roll the disc carefully to avoid slippage to mark out R. Job done. I construct the circle using this new measured portion.

But what is this measurement 2π ? By constructing a equilateral triangle base R and marking of the distance r we have a ratio comparison equivalent to division ( it is the geometrical basis of division). From this i can see that 2π is slightly more than 6*r but a good approximation is given by 3 : 19. I obtain this ratio by carefully "dividing" the small difference, and marking it off carefully on the other side of the triangle.

This process of approximation of the measure for the ratio can be repeated for further accuracy, thus highlighting the measure as the value of the division process, not a number!

Of course the numeral namespace has been constructed to be able to name these measurements.

Thus i think the measure line concept along with ratios provides a foundational basis for ordering and evaluating the world and with a namespace we can describe that evaluation.

In addition the measure line is a flexible bur inelastic measure and so can measure curves, depending on the degrees of freedom given between the fractal regional boundaries as the iterations are increased. Thus a tailor's tape is an adequate measure line for most discussions.

Cool! May the muse inspire you! :music:

So i fell to wondering how the proto-Akkadians as early Sumerians and Babylonians were able to divide the circle into 360.


I remember Euclidean constructions and mistakenly thought that this was the sum of their technique back then. Thus the issue was trisecting and quntisecting (http://www.jimloy.com/geometry/trisect.htm#tools) an angle. This i thought must have been easy. In fact it is not that hard for a practical person. This is when i found out about Galois theory and the greek "game" or agonia of trying to do this with an unmarked ruler and a pair of compasses!

so i fell under that spell for a while and diverted much of my thought and time to impossible pursuits! Why would you?

This is precisely where Geometry and thus a whole swathe of mathematics becomes obscure and agonising! The greek love of the Agonia!

Like all things the game is a diversion that produces amazing and interesting responses, but ultimately is not a model of "reality", by which i mean many of us have had our world view twisted by this type of practice and constraint. It is also why i decided to coin the term spaciometry to reveal in myself these imposed twists by adopting a different viewpoint.

So enjoy the pursuit as much as anything else but the fundamental connection does not require one to undergo "agonia", or any other "initiation" rite into the secret society of the pythagoreans etc.

Now this is out in the open, my appreciation of the vortex and its surface manifestation as a spiral has been enhanced, and the contention that the circle is a special spiral has an increased confidence value. The fundamental motions in the spaciometric motion field are spaciometric rotation and spaciometric extension. These mean that the fundamental form is the vortex/spiral (torus-vortex in 3d). If i have a condition that fixes the extension to a constant then the special form of a spiral circle or sphere is the result. Thus the extension quantity and its ratio to the rotation quantity determine all forms from the so called singularity to the infinite "straight" edge.

The proto-Akkadians are linked by trade and via the Dravidians to the Chinese ancestors. There is also a land bridge between them admittedly over difficult terrain. The Chinese have a construction of the i Ching based on 24 divisions of the 6 concentric circles (http://www.chinesefortunecalendar.com/yinyang.htm). This tool enables the practical division of any circle into 360 parts, by qunitisection, and bisection and trisection.

Lai Zhide is credited with introducing the Taijitsu called the Yin Yang, and in particular contributing to Chinese "analogical thinking" or philosophy . One must not get hung up on the apparent absurdity of Chinese astrology, as it is in fact an ancient and well respected body of knowledge which is common in the west and indeed Astrology  gave birth to western Astronomy and Cosmology!

It is a thing of note that the Babylonian magi began to demonstrate philosophical thinking, dealing with abstractions and principles in their cuneiform tablet records early in our common history. Chinese Astrology is nothing if it is not an exact example of this ancient "scientific" analysis of the cosmos.

Analogical thinking requires a form to make the analogy with. In chinese this is called the Yi. Thus the five elements are just 5 Yi by which chinese philosophy assembled the chemical and physical properties of the cosmos. They represent in one analogy a kind of periodic table, and in another a type of electromagneto hydrodynamic spectrum. The simplicity of analogical thinking is that it can be ratioed to just about anything.

In this current western absolutist, abstract phase we find it hard to conceive of the usefulness of this conception in terms of dealing with information overload. In terms of creativity, it is not at all restrictive or stifling, nor does it preclude innovation. Chinese scientists are as objective or subjective as i and i require an organising principle or set of principles to store knowledge!

Lai Zhide i think would have been at home with Carl Sagan, Richard Feynman even Einstein and Newton in his philosophical reduction of the cosmos to the i ching and his consumate Yi-ology.


(http://www.fractalforums.com/gallery/3/2367_23_07_10_4_42_40.jpeg)

There is a recurring relationship between spirals vortices and the measure 3. this in fact could be π 0r e but this certainly has a spaciometic form which is irregular angular and cone or pyramid-like. This has a bearing on close packing in the sense that a spiral based around a 3 measure centre may provide the best bundling spaciometrically. Certainly i notice the measure 3 in a lot of natural plant forms and structures.

I thought it might be worth exploring the notion of extension spaciometrically, as the notion of straight and right are cultural paradigms, and somewhat idealised. I will consider this more fully after i have completed a survey of kinesis and kinesthesia with its subfield proprioception.

The Logos Response is informed and grounded in my sensor systems and processing, and thus extension is not as simple as it is made out to be.

I rather suspect that spiral motion will be he outcome and that orthogonality will be important along with orientation and feedforward feedback cybernetic systems.

It is of interest that orthogonality is a distinct sensor system of itself, irrespective of any spaciometric definition. Orthogonality founds nd contributes to spaciometry, and in a very real sense spaciometry could not exist without it.

By the way i feel a clear distinction between humans and other animates is spaciometry as an axiomatic system. Other animates i am sure have spaciometries and even codify and pass them on to the next generation, but it seems only humans seek principles on the basis of analogy, and reconstruct there experiential continuum accordingly.

That is not to say that other animates do not develop erroneous concepts of the environment, cos clearly they do, but there information and processing is empirical not axiomatic. I guess in just about everything else we are similar including inductive and deductive reasoning!

The notion of relative ratioed radial expansion suggests itself as the fundamental motion with relative ratioed radial extension being the specific or abstracted orientation. Thus extension and orientation are the exact same notion. The ratio is specific to the form and or region of focus, but the relativity relates to the quantifying of expansion by ratioed extension in a standard orientation.

If the radius is special, that is inelastic it produces a circular expansion and the notion of perfect or universal applicability of the measure: a universal metric. However if the radius is "elastic" or deformable in some way then there is no universal metric and all measurement of extension is local and dependent on orientation. This would be the case if a spiral reference framework is utilised.

Vorticular motion fields may indicate that the underlying metric is based on a ratioed radial expansion that is "elastic" relative to orientation.
Dilative rotation (http://www.matematicasvisuales.com/english/html/geometry/transformaciones/rotdilat.html)

Polar coordinates, spherical coordinates (http://en.wikipedia.org/wiki/Polar_coordinate_system) are the most natural relativistic reference framework currently in use, but it is still made out to be special, whereas it is due to historical primacy that Cartesian coordinates are still presented first. The weight of mathematical description is still presented as Cartesian, thus Special Relativity seems inaccessible and strange, and the "complex plane" can pass as a number system rather than the coordinate system which it is.

Spiral coordinates will be a truely strange system, but one i am laying the groundwork to explore.

Plato was one of the first to discuss the problems of perspective. "Thus (through perspective) every sort of confusion is revealed within us; and this is that weakness of the human mind on which the art of conjuring and of deceiving by light and shadow and other ingenious devices imposes, having an effect upon us like magic... And the arts of measuring and numbering and weighing come to the rescue of the human understanding—there is the beauty of them—and the apparent greater or less, or more or heavier, no longer have the mastery over us, but give way before calculation and measure and weight?"[15]

from Perspective (graphical)
From Wikipedia, the free encyclopedia

To which i add the art of languaging, without which the full context of the other three is missed and the logos response- the calculation-takes abnormal preeminence.

By the way only the ascetics among humans describe the human mind in these pejorative terms: a confused "mind" is no weaker or stronger than a seemingly understanding "mind" it is merely confused. The actions proceeding from a confused "mind" may appear weak and ineffectual but appearances can be deceiving, as outomes determine the relevant value of states of mind. And in any case many confused 'minds" have taken decisive and bold action without understanding, just as understanding "minds" have taken weak and indecisive action.

We have no real sense of "time" that is not spiral, thus our modern abstract notions of "Time" are simply deceptive, hiding (http://members.aon.at/sundials/gnomone.htm) the creeping advance of precessional motion and indeed all vorticular motion in the void. Instead of shadows we replace our connection to our universe with something more insubstantial: the notion of time.

The eyes have it (http://www.theregister.co.uk/2010/09/16/vision_superchip/) for more go here (http://www.eng.yale.edu/elab/research/svision/svision.html)

The important point is  it is not the technology it is the cybernetic system, based on iterative processing called here convolution.

It's nice you're keeping this thread going.  It's enjoyable to wander among your thoughts occasionally. 

It seems that projective geometry, in particular mapping projections like mercator et al may be a way to enscribe vorticular spiral wraps onto any object, thus providing a transform from its usual geometrical representation to a sonic wrap. This spaciometric transform means thqat in essence i would be able to represent all objects by these wraps and deal with the motion laws for these wraps for a theory of everything.

Somewhere nice and spaciometric to play (http://jamaicalive.com.jm/?q=ElicaTeam&video=dMRUXTUDyx8)

It strikes me this morning that the measuring instrument that solves all my problems is the pedometer. No not all! my problems,just the curious reversal tht has taken place between the spiral and the circle.

To whit, using a pedometer where the circumference of the wheel is the unit of length allow everything to be measured in radians. In particular every curve from a straight line to a circle can be measured simply and easily by pedometer. Screw the greek straight rule without any marks i say! ;D.

So then the straight line and the circle become two special bounding curves for every other curve, in that every other curve is spiral and tending toward a straight line or a circle, no matter how many twists and turns or knots and winds it gets itself into.
 I could go further and define each curve as a unique spaciometry, to be explored, and thus categorize spaciometries by the fundamental curve used in their constructions. Thus a circular spaciometry and a straight line spaciometry would be 2 special spaciometries, and more general spaciometries would be spiral.

Releasing the measurement method from the spaciometry, or rather restricting it to the spaciometry that can accommodate all others allows me to apply one single fractal pattern to all measurements and inherently simplifies things.

This does not require a new aggregation base as it is a scalar of the old . so it is not (mod π^n) based but π*mod(10^n).

In practice then if i measure a length 3π it is up to me to specify the curve that i am measuring along. The curve itself informs me whether it is a rotation or a straight line or a combination. Angle measure becomes plainly a tool utilised to describe an aspect of a form and may be generalised by the measuring curve used in the tool. thus angle measure does not have to be exclusively a circle anymore, but it does have to be a specified and constructible boundary curve element.

using the pedometer measure it is straightforward to mark off a radian: it is the distance the pedometer rolls equal to a radius of the pedometer. So using my pizza cutter for measuring radians is quite fun. It is clear that if you roll along a straight line that the distance in radians is simply how many radii divide the length in normal units, but for a curve the distance in radians has to be read off from the pedometer. This makes it clear that for a curve a judgement of the lIne is taking place, and in fact in general the curve will be measured by an approximation to its shape.

This makes every curve measure potentially a bezier polynomial, and polynomials in general the family of curves for approximating a general curve and a general curve length. Therefore polynomial interpolation and extrapolation remain the fundamental basis for measurement, and the nature of calculus is preserved and revealed as polynomial approximation to curve shape, curve length and curvature itself, based on a spaciometric notion of dividing off a standard curve of best fit for each division of the focus curve.

In a straightforward approach this leads to the binomial series as newton discovered, but in general to the Taylor series and the special case the maclaurin series.

As much as these are presented as results of differentiation they are no different to dividing off the curve and seeing which polynomial best fits. The use of the binomial theorem and the binomial coefficients is one of those wonderful things that occur in investigating the set notFS,the universal application of an algorithm.

The recipe for finding the best fit is the same as the recipe for organizing patterns of medic rhyme and the organizing of finite sets of things. So if we want to choose from the possible permutations of a finite or ordered set it is not unreasonable to try and organise it by the binomial expansion. What is sup rising is how well it works apparently.

I say apparently because it is an algorithm, which essentially means an iterative set of instructions on how to tackle the problem in general. A valid algorithm will produce valid results. Two valid algorithms may not produce the same looking results by form but by a common measure the assigned value usually converge.

However sometimes the results of valid algorithms diverge which begs an interesting question? How is validity assigned?

The exploration of this strikes at the common notion of proof, and in fact in general I adopt an evidential basis for a theory not a so called "proof" basis. Falsifiability as Karl popper states is in fact the only occurs razor we have. I use it all the time to review my ideas and welcome a falsification by evidence as a new avenue of exploration and play

So this morning it becomes clear that the tools and customs we use are so embedded as to make it seem fanciful to use any others. However as far as rotation is concerned i may use any specified tool . Thus degrees are merely a fractal pattern marked on a circular tool(protractor), and radians are a different fractal pattern marked usually on a unit circle. The measure line concept is a fractal pattern marked on a line (rule) and the line  is said to be "straight", but this is an undefined concept. However we all agree more or less what "straight" is, but again this is a cultural definition. Some attempts to tautologically define straight are usually made .

Thus it is clear that to be rigorous one needs to define the fractal pattern and the "curve" used in the measuring tool, and the application of the tool for its measuring purpose. This in essence is a simple algebra, or rather the axioms of the algebra of the measuring tool. Changing the measuring tool changes the algebra, but in most cases only by a scale factor. the measuring tool can be exotic in curve and shape, so a spiral form may seem a little tame after a while!

Weights and "forces" represent a different measure principle, as a fractal pattern is used only after the motion has been transduced to a form that can be measured by a fractal "Scale". This scale is arrived at by dimensioning the motion and then parameterising the dimension. A parameter that has a tractable or useable uniformity is the preferred choice for a scale, but again this is a cultural decision. The transducing of the motion by dimensioning and parameterisation is the attributing of measurable properties to the motion, and the establishing of a relativistic reference frame. Relativistic motion and relativistic motion transfer follow from this transduction, but independent of the reference frame we observe motion and motion interactions.

It is to be noted that dimensioning is governed by the measuring tool. Thus we dimension on a cartesion coordinate basis because we tend to mark the boundary of objects by straight lines. With a minimum of 3 distinctions we can record the boundary of an object if it is regular. Irregular objects require many more dimensions. The number of dimensions also varies with the meassuring tool. A tool that fits the form typically requires fewer dimensions.

Spiders are a friendly folk imparting wisdom as they go, spinning yarns of interminable length and spiraling in the wind.

So i noticed hat a spider does not build a spiral on circular elements, but on triangular and radiating elements.  (http://www.brisbaneinsects.com/brisbane_weavers/SpiderWeb.htm)

The spider seems to typically lay down or attach a main anchor line. This is so fine and uner tension so it is the nearest thing to a straight line unaffected by gravity, and interacting with the aerodynamic forces that counteract any gravitational ones. The spider then drops at least 2 more lines to provide a stable platform to anchor the radiating spokes of its web. Thus it uses radiating lines a the basis of its spiral . Radiating lines are interesting.

I probably first came across radiating lines as a pattern due to sunlight passing through my eye lashes, and then spokes in all sorts of situations from basketwork, sowing patterns , wheel spokes, spider webs.
Radiation is an attribute of straight lines originating from a single point and spaciometrically rotating a boundary. It is a structure that has spaciometric mass and spaciometric density which varies with relative radial expansion from the origin.  

Spiders typically lay out between 28 qnd 34 of these radiating lines before spiralling. Thus a circle does not seem to preoccupy a spider's mind, simply a centre point from which to radiate to any given boundary, preferably stable.

Radial expansion then is a spaciometric concept based on spaciometric orientation and spaciometric rotation of a boundary from a point of origin or "centre". This is a concept built on iterative processes, and my perception of radiating lines is an iterative process.

Relative radial expansion therefore is a process that describes the general radial motion in motion space and spaciometric rotation is a process that describes the general motion in the same space that is not solely radial.

Spaciometrically i would want to reduce motion to these 2  fundamental types: radial and rotational. All motion are then some combination of these fundamentals.

I suspect my spider friends have more to teach me about spirals, especially spiral wraps. (http://www.brisbaneinsects.com/brisbane_weavers/RTent.htm)

As i have recently added to the Axioms, it occurs that relativistic motion attributes of solidity, liquidity and gaseousness relate to the density and vigorousness of spiral motion and the direction of spin. Thus a hot gas has a higher spaciometric density of helical vortices all spinning in the same way which would tend to make hem energetically bounce off each other and spread apart in all directions. A cooler boundary condition would tend to absorb this spin and start a counterclockwise spin. This would tend to keep the helical vortex motions fluidly linked. Finally a cold region would tend to have very slow rotation and therefore the regions would tend to stay together in viscous lattice forms, with slow thermodynamically generated spins. Nevertheless the electromagneto hydrodynamic spin would then take precedence giving rise to electro mechanical phase attributes.

Often the philosophy of the scientist is squeezed out of the description, but it is all important at the level of observation. What one thinks determines what one perceives (http://docs.google.com/viewer?a=v&q=cache:3n-yv-grxKkJ:www.dpedtech.com/NegVel.pdf+spiral+reference+frames&hl=en&pid=bl&srcid=ADGEESjUCBLFIydo4cE31HsMg_-uDYV7R2iQN43_gVR8rEZt332zNv5j7D_DwfmV9S7nfeqityuYJOfTEnH0sxul1rsdAb49BH5DyGjUjRv_gY_IULyNU0xQ4c9AWWl4mdmJ4FRe1K7G&sig=AHIEtbRdyNKyY3Wsj9kOmgovd9DRNsmiSw)

I am thinking about the foundations of mathematics and what i have explored so far. So I feel i will move toward a tidying up and consolidation of where i have got to soon. Fractions i feel are the only outstanding thing to explore and Abstract Geometry.

I am not surprised to find that mathematics is not solely a human animal affair, and that aspects of it exist from viral through microbial to zoological forms of life. The simple description is everything is mathematics and nothing is mathematical!

I was privileged to see my spider friends constructing their spirals in the early morning sunlight!

As Geometers go, they ain't half bad! They measure, they locate, they mark and they construct, and they use their whole body to do all of this because it is a survival technique that engages them in the panoply of life.

Even the spider shows the fundamental importance of radials and rotations. If i have observed anything of significance in this exploration it is the Logos Response and that the fundamental reference framework is Radials, and the fundamental motion is Rotation and Radial Expansion and the fundamental form is Vortex-Torus, and this is fundamentally because their is no constant but Constant Change.

That's life, as they say, and mathematics has been an organisms response to life from inception, and will continue to grow and develop with organisms as they respond to constant change. Mathematics therefore is inate and intrinsically simple and intuitive, and anyone or anything that makes it not so is perhaps not really a mathematician.

In fact, get rid! I do not want mathematicians anymore i want craftsmen, artists, artisans, builders, potters, weavers, jewelers beaders, shapers who have a deep love of what they do and who study and play with what they love to make it more utilitarian, more enjoyable for themselves. That is where mathematics is born, nurtured and thrives.

The i ching algorithm and the Lai Zhide Taijitu or yin Yang symbol are a result of a common principle which applies to rotating reference frames. Essentially Radials and rotation will produce similarity in form  (http://wn.com/the_interplanetary_magnetic_field_parker_spiral)   irrespective of the topic: mathematics, science art astronomy astrology....

Brownian motion (http://en.wikipedia.org/wiki/Brownian_motion) and Bachelier theory applied to stock markets get heat from Benoit! Very stochastic!

http://www.youtube.com/watch?v=vxbxXBrOPS8&NR=1

What interests me is the non linear time scale. Very iterative,non intuitive and fractal self similarity.

Fractions  (http://nrich.maths.org/2515) and fractals  before breakfast! 2 impossible things, eh? ;D

Some foundational (http://www.pballew.net/arithme1.html) words and where they come from.
Thank 'h¶h for the Brahmins! Fractions (http://archives.math.utk.edu/articles/atuyl/confrac/history.html) would be a piece of cake otherwise!
Romans of course weighed out their portions (http://en.wikipedia.org/wiki/Roman_numerals#Fractions). Give them an uncia and they would take a Roman Legion!

So lets throw in some polynomial thinking and see what we get (http://sites.google.com/site/scottyuancosmos2009/home/continued-fraction-expansion-of-a-real-number). Continuous fractions!

Now we can see the Egyptians in a new light (http://historyofegyptianfractions.blogspot.com/), with some nice hieroglyphs (http://euler.slu.edu/escher/index.php/History_and_Numbers).

Oh the Humanity! Oh the irrationality (http://en.wikipedia.org/wiki/Irrational_number) of it all! The magnitude (http://en.wikipedia.org/wiki/Magnitude_(mathematics)) of these ratios and incommensurables is a significant clue to the measure line concept.

By this means of education and cross fertilisation and promulgation of conceptions Fractions morphed from portions, and ratios into magnitudes and numbers: and i propose that number morph to the concept of measure as a spaciometric entity of dimension, with wholeness linked to boundary and fractal pattern self similarity.

Fractions are "always" (http://docs.google.com/viewer?a=v&q=cache:AGmdNBykvIAJ:www.dataweb.nl/~cool/Papers/Math/ProportionsAndFractions.pdf+Fractions+origins+number+line+proportion&hl=en&pid=bl&srcid=ADGEEShYIl0TCvZzhy6E-xM1kiJwx4izT6mUdRd4yVDs80lzAhNmbgPYuiA937PRlt0cl0zw23tW1iENARSFeetaCGtKUqyLwMNC_gTUS7CDKcqxv9OSZvdS1hi6ONRSZueOHDRcBd9g&sig=AHIEtbTjZ_kjPu7oA_XTeXTepEJi2lFhaA) presented bereft of their ratio origins. And even as Rational (http://en.wikipedia.org/wiki/Rational_number) numbers the ratio geometrical thinking of the earlier mathematicians is downplayed. Ratios and there cognates proportion and portion are crucial to an intuitive and spaciometric understanding of measure and quantity and rank and enumeration etc. The notion of Tensor is accessible through these fundamental notions of ratio and proportion (http://en.wikipedia.org/wiki/Eudoxus_of_Cnidus) and portion. So good on yer! Eudoxus, and any other ancient mathematicians who valued proportion. And Archimedian properties are attributable to Eudoxus it seems. Archimedes always was a good student!

Enjoy the reading. It's Intense! ;D

The google sketchup (http://sketchup.google.com/intl/en/training/videos/new_to_gsu.html) tool/ app is a sufficiently complex and sophisticated Representational system for me to use it as an instance of what i mean in part by spaciometry.

Despite its complexity it is intuitive and easy to learn and engaging.

With a tool like this i can re-explain notions of parameter, dimension, radials and spaciometric rotation, spaciometric mass, density and surface form and boundarisation. I can also emphasise the tool nature of many fundamental geometrical "givens" and the abstract nature of Euclidian, Greek Geometry. I can illustrate Riemannian geometry and how restrictive Euclidean geometry is compared to it, but how even Riemannian geometry has to give way to a more general spaciometry.

This fundamental tool is not new, but new use can be made of it in redefining the foundations of mathematics.

   Mathematics derives from the french mathematique which derives back to the indo European roots through Latin and Greek and the Sanskrit: man.

So in every sense mathematics is essential to the defining of man or what it means to be a human. This does not imply arrogance ut rather a time when animals became self aware enough to describe there learning and thinking ability as "man". From this one distinction flows every human notion of difference to other animals.

However the Root of interest for the Logos Response is: me. Now this distinction is what our CNS and PNS do prior to and during man (that is learning and thinking) and that is MEAsuring and comparing.
Thus me and man are possibly the 2 oldest "human" notions, but are not unique to humans as all animals and even plants and fungi do this. It is a fundamental response of life (living things).

From this root analysis no human is excluded from mathematics as it is part of ones definition of being human!  And every human not only thinks and learns but also measures and compares. Thus the problem is the distinctions that are drawn, or posited and adopted. These are what set up and establish barriers as well as boundaries between the myriad human activities that would otherwise qualify to be called mathematical!
So be proud and loud you craftsmen and craftswomen, for everything your hand produces from your mind is mathematical through and through, especially if done with love:

Mind \Mind\ (m[imac]nd), n. [AS. mynd, gemynd; akin to OHG.
   minna memory, love, G. minne love, Dan. minde mind, memory,
   remembrance, consent, vote, Sw. minne memory, Icel. minni,
   Goth. gamunds, L. mens, mentis, mind, Gr. me`nos, Skr. manas
   mind, man to think. [root]104, 278. Cf. Comment, Man,
   Mean, v., 3d Mental, Mignonette, Minion, Mnemonic,
   Money.]

From Free dictionary.

From this it is but a short step to........play!

The establishment of distinctions is not only a language response, but also drives communication, explanation, education and differentiation. It is a powerful aspect of the Logos Response which accounts for many communication motivations and road blocks.

However just as thinking and learning measuring and comparing are at the heart of being human, so my friends is doing math, or rather playing with the things around one and thinking and learning about what that reveals.

Given the derivation roots man and me can anyone suggest a more friendly name for mathematics?

http://www.youtube.com/watch?v=BQRIIeMwfR0&NR=1

On the derivation of measure, and play.
I just found it.

Experiment in proprioception 1
Slowly, very slowly turn, or twist with eyes closed.

There is a quantum size to sensors action potential. So i know i am moving by 1 intention, 2 lateral differences between proprioceptors, 3 mesh of proprioceptors self referencing, 4 exterioceptors (open eyes) audiometric and gustatory differentials 5, memory comparisons that is  perception comparisons.

Each one in this list requires an increased potential to activate.

can i get some of what you're smoking?  O0

:o

When i generalise the notion of a coordinate system to have n ordinates/ axis not necessarily mutually orthogonal, which in any case is a non sequitur beyond 3, i find myself tracing a path through all the ordinates ending on the point i am referencing. Thus it seems to me that a generalisation of a coordinate bracket would be a path bracket, based on a sequence of visiting the ordinates.


Thus it seems fair to think of a peano curve as the basis of a sequence for visiting each ordinate on an extremely dense ordinate system.

Experiment in proprioception 2

Pick something you know how to do well. Do this as fast as you can. Repeat and attempt to go faster. If you can repeat faster still.

Your consciousness of doing this task is altered. You will not know how each movement flows into the next, because your attentional focus is limited.

You routinely do more through unconscious proprioception than you can distinguish.

@ lycium  :D Hey man if you want weed, you can smoke weed, but if you want what i am on you need to spin out, dude--right to the edge and beyond!  ;D

So now the spiral gives up another secret.

The circle and the spiral are in competition in my mind. Well not really as i think that the spiral is prior to the circle, and in fact i have gone a long way to generalising the boundary properties for generating spirals. So it is really a bit of penile envy. Why is the circle so damn easy to use to explain spiral motion in general?

The circle is a very very special spiral form, and i have yet to rigorously determine it as an intersection form for the general spiral boundary. It can't be because the general spiral form is chaotic, so why do circles introduce an order to that chaos? Why are they so facile as a starting point for anything spiral?

A sphere or a circle can enclose any shape, and then the boundary of the shape shrunk to, thus providing an increasingly accurate map based on the radial adjustments to a circle or a sphere. This means i can develop a theory of interactions based on circles or spheres and then apply it to real objects by using a radial transform map.

Is the theory based on the circle or sphere more general than the one based on the actual shape of things? Yes, but that makes it an approximation which has to be made more applicable by minute adjustments.

Well i have heard of this type of limit process before, and it in fact underlies the notions of differential geometries. So even in our most general thinking the circle plays a powerful role, and there is no obvious sign of the ubiquitous spiral or torus forms, In fact we tend to explore these form using the circle!

And i have done the same.

I was thinking about trochoids and cardioids in their relation to the Lai Zhide Torus based on the I Ching algorithm. I wondered what the path of points on an unrolling spiral would look like. It turns out to be trivial in one sense but non trivia in another as the programmes that can generate an "unrolling carpet" animation are non trivial.

It is trivial because the answer is that they form a set of trochoids (http://books.google.com/books?id=STDYCmXrrE0C&pg=PA288&lpg=PA288&dq=trochoidal+curve+spiral+unroll&source=bl&ots=_vBv9DuzUn&sig=vUc7hrsWFL2uXoWdMZv6o9waKSo&hl=en&ei=NDq5TMv1M4m9jAf53ZnIDg&sa=X&oi=book_result&ct=result&resnum=1&ved=0CBEQ6AEwAA#v=onepage&q&f=false) originating on the line along which the spiral is rolling. It is non trivial in that the trochoids (http://linuxgazette.net/133/luana.html) dimensions are radially decreasing to zero for unrolling, and increasing for rolling.

Thus i can see the straight line and the trochoids are geared (http://en.wikipedia.org/wiki/List_of_gear_nomenclature) to the spiral and the circle is inextricably enmeshed. I can of course transform to a different boundary say a spirangle but that only emphasises the general applicability of the circle in solving these types of problems.

A general geometry for the trochoids is found here (http://www.archive.org/stream/atreatiseoncycl00procgoog/atreatiseoncycl00procgoog_djvu.txt) with some illustrations here (http://chestofbooks.com/crafts/children/Paper-Folding/XIV-Miscellaneous-Curves-Continued.html)

The usefulness (http://www.scientific.net/AMR.24-25.149) of these curves is probably an industrial open secret but even toys inspire (http://docs.google.com/viewer?a=v&q=cache:555KrfRVqSMJ:poncelet.math.nthu.edu.tw/disk5/js/cardioid/6.pdf+trochoidal+curve+spiral+unroll&hl=en&pid=bl&srcid=ADGEESg8mf0u0bE0InayWl_oKy7vzL4t3ZGjMXuz7BE4dFMPs3gLvtDcuO21r7SQr-gT0Hv5prGdzZLdKIpK2k1AsuOyBQeSRFUsriv6jyzp0fkVMSyKQsG1j1rbmSGKtjL6hAnPZDcK&sig=AHIEtbQedJUVVTgGOApkBndkNAcGqo4mQA) this deep use.

However if only for the Quote at the end i like this exploration into thr problems of locomotion on roads (http://docs.google.com/viewer?a=v&q=cache:Hh1Gd-1vxw0J:www.maa.org/pubs/sampMMA.pdf+trochoidal+curve+spiral+unroll&hl=en&pid=bl&srcid=ADGEESiY4VifrYMx20VIym_9o7dUttwD12Apyi6rrDyd83OSXGVHqekC-Snb4DM7jx-dv2ovK_JOt-oPfo9IO_h7QWN7fhJMIKJC68Cl9VpZPsRg-HGRMu2kLkmrPnVkYi_rAjqGfH7V&sig=AHIEtbRqagTeFMnlBQY0SAVPKfkpgl4-sw).

I always enjoy a good blow (http://docs.google.com/viewer?a=v&q=cache:oq_kVkYtaaMJ:journals.ametsoc.org/doi/pdf/10.1175/1520-0469(1980)037%253C0930%253AATDTCM%253E2.0.CO%253B2+trochoidal+curve+spiral+unroll&hl=en&pid=bl&srcid=ADGEESjyrq41djyU8K_5YTG4v57R9cerAGQxGz-vYUCVGqwZZHVK-Dh_nqxvFREn27-byGnDgFwSpRcPiKLXiUzBH5JOe7Glf_U5CM5BctTCFMxjNktqrc7icjYUzKtIdKkA2wqBwuYH&sig=AHIEtbT1MMAmnSUepe-YSHa6ur4ErYl_-w) in 3 dimensions! but for me the simple bouncing of a ball before it comes to rest hopefully before smashing that window is illustration enough of the trochoidal nature of movement and motion and the interlinking of spiral and circle, of vortex and torus.

The circle is a very special form of spiral as the torus is a very special form of vortex. I do not think spheres actually are real objects,but ovoids are which are a form of torus approaching zero central radius.

The power of the circle or sphere lies in its unique abstraction from every vorticular form, and that is the sense in which i think it is an intersection of vorticular or spiral forms unique to animate consciousness.

Ah yes, i forgot to add that i think the mandelbrot set is intimately related to the trochoids of an unravelling system of spirals in the polar coordinate plane, and consequently the mandelbulb is intimately related to the three dimensional trochoid shells unravelling from a system of tori (toruses!) cut off, and ultimately shaped by the circular or spherical boundary condition.

Proprioception experiment 3:

Go into a familiar room and stand in a familiar space. Now reach behind you to grab, or make contact with something.

Notice how proprioception is tied in with visualisation, and how you use the visualisation and the audiometric senses to locate and position your hand and body to ensure maximum success.

If you are struggling, try to smell out the object or surface or use your Whiskers (small hairs on your body) to sense how close you are. Maybe a temperature feedback might help, but these are all exterioceptors, so realise how exterioception assists and coordinates with proprioception.

And you thought geometry was just Euclid! :dink:

i have commented on the notion of dimension. in trems of the characteristic 3 d and 2d nomenclature. So essentially the idea is that a 4th orthogonal dimension exists, etc. Of course a 3 d consciousness cannot perceive a 4d spatiality, so the rubric goes.

Well i will be clear and perhaps reactionary in saying i do not think an mutually  orthogonal direction beyond 3 exists.

I think that there are lots of dimensions, but orthogonality is a unique characteristic to 3d. That is to say to flatlanders that they would not perceive orthogonality in their flat world, only us 3d types can see it in their world and in ours.
 
Now flatlanders is a "teaching myth", and thus it is not an analogy that i would care to continue beyond 3d,because in 3d we have all the dimensions we wish.

So why is orthogonality important? Orthogonality is in fact a sensor driven notion, and we have an orthogonal sensor within the vestibular system . That for me locates it in what we call 3d. Now if we had a 4d orthogonal sensor i would concur, but we do not.

What we do have is a proprioceptive mesh network which drives our motion of geometry,and our imagination. Thus proprioceptively we can imagine another dimension, but in the sense of a "degree of freedom" . We do not like to be stuck and 3d is so often defined in that sense of a limit to our motion. Of course mechanics and engineers know that there are degrees of freedom and the rigidity implied by 3d is a difficult thing to achieve without taking these into account.

So while it is a nice story and all that the 4d fiction is actually masking what is going on. We are actually dimensioning in terms of degrees of freedom.

Thus if i build a so called hypercube "shadow" (http://www.miqel.com/fractals_math_patterns/visual-math-hyper-dimensional.html) i am actually building a shape that with an extra degree of freedom providing it is an elastic or compressive degree of freedom can deform while preserving local relationships. As a consequence like any of my real world clothes seen as mesh networks i can turn them inside out continually by rotating in the correct way. I can also twist them and deform them in other ways while maintaining relative relationships.

Some of the more complex web like structures can still do this but in a complex internal radiating in/out pattern which in fact may be good models for growth and development within a 3d cellular structure.

So are my clothes shadows of 4d objects? You know when you get that feeling you are not lone.... :dink:

Just a note to myself. Trochoids and torii are related in some way. Maybe that torus is a 3d trochoid mesh, and this links to helical wraps of torii, maybe they are types of trochoid in 3d. Now i have seen one description of the lie group as the symmetry of 248 circles (http://www.youtube.com/watch?v=pCDxsBmaXU0) arrange in an orthogonal twisted like pattern so maybe trochoids relate to Lie groups in some fundamental way.

Benfords Law (http://www.newscientist.com/article/mg20827824.700-curious-mathematical-law-is-rife-in-nature.html), Eudoxus and Archimedes are related i think. We typically do not use the full range of any aggregate number system, preferring to rescale. This kind of hides the infinite fractal nature of things, but also relates to computational boundaries being a scale issue. That means that my PNS and CNS based massively parallel distributed computing system computes surfaces based on scale information. Therefore a smooth surface at one scale becomes a rocky mountainous terrain at a high magnification because the scale information falls within the computational accuracy limits.
In one sense computational error is what i live with and in everyday, so notFS may be a scale issue as well as a sensor issue. Who knows what i am missing because of scale or not having the right sensors?!

It's a bit scary really, cos i am blind and deaf and insensate in so many areas! Or its exciting news- plenty to explore and find out for everyone!

I want to extend the conic section curves to include loxodromes and trochoids. The way i would do it is by projecting the poxodromes onto the cone that is enclosing andtangential to the sphere. Similarly i would project the trochoids onto the cone.

All these additional conic section curves would then be the defining set of curved motion for gravity.

Additionally i would project thr trochoids and loxodromes on a torus onto the cone. The torus would again be enclosed and tangential to the cone.

With this thought in mind i would project the curves of a lie group onto a cone and search for matches with the conic section curves. For the lie group to model gravity it would have to match precisely with all the conic curves including the extensions. 

Topology, i say, is the analysis of the the properties of the space that inheres within a form. And when i say propeties i mean attributes to that space.

So i attribute a property to a space and maintain that attribute by some means and explore the consequences of that attribution to that space by any specific or general action.

This is as general as i want to get because i feel that the mathematical definitions of topology are way to abstract to be graspable. I have used form property and agent to define topology so that i can locate the subject in reality. One of the reasons why i escaped to spaciometry was so that i could get clear of all the abstractions and words. It seems that mathematical definitions are more about words than anything tangible, whereas i know every mathematical idea comes from a tangible source, but some would want to make out otherwise.

Topology naturally includes any geometry and is an interface between geometries in the abstract and the material arts and sciences.

So a concrete example: an elastic band inheres a space to which i attribute elasticity as a property. I manitain that property by playing with loads of elastic bands and referring to a cultural database on elastic bands, which informs my exploration and apprehension f the property of elasticity.
One of my explorations for example may be what happens to to marks on the elastic band as i act on it by stretching twisting folding. A question arises? what happens when the band breaks? This is of course explored in material sciences and thus the tw studies mutually inform each other.

So topology is not an abstract set of words and notations but an exploration of real spaces.

With this in mind i look at the property of "tangential". In reality this does not exist.

A surface of one form grazing another surface does exist. In practice we deal with tangentiality by using a variable area of contact that can be made arbitrarily small.

In fact it is a variable volume of contact, and the arbitrary measure of closeness has to be suitably defined. The tangent is then defined as the limit of a process that makes this volume tend to zero. Although this is defined as a point in practice it is never achieved as the materials breaks free of each other before this singleton point is ever reached. The reason is quite simple: a singleton point is a fiction that does not exist. The consequence of this is that tangents do not exist as defined normally.

This leads to a point about points in Euclidean geometries. They do not exist. Instead intersection of lines is and always has been used. The introduction of the notion of point into Euclidean geometry is a later addittion to try to axiomatize the system. Since it has no practical geometrical purpose that intersection does nor fulfill, point has not been analysed that much. However i say that it does not bear scrutiny and its legacy in mathematics has always been troublesome.

Manipume‘.

That's my first idea for a new name for mathematics based on man and me the indo european roots and manipulate which itself has an interesting etymology.  The idea is :think, manipulate, measure.

I really like the quote below so i have attached it.

Rolling a carpet produces radial order from linear order while maintaining linear order.

Rolling a carpet enables a linear sequence to connect to sequences that appear radially in a unique way for each radial sequence.

Rolling a carpet takes a linear, sequenced order of regions and connects them radially to each other in a radial linear sequenced order, but the motion to connect them is  trochoidal and the combined motion of these  spiral trochoids may appear chaotic.

Rolling is one way of ordering chaotic motion.

Rolling a ball of wool produces radial order from linear sequence order in 3d while maintaining linear sequence order.

The motion of the regions in the linear sequence order may define a 3d trochoidal type.

Trochoids are possibly the curves of creation, by which i mean the curves involved in every motion that creates order in equilibrium. They may also be the curves of destruction, that is the motion curves of the breaking up of an ordered system through dis-equilibrium.

Trochoids especially spiral trochoids i think are involved in torus formation, which means in the formation of general forms from bubbles to clouds, and in the formation of plants like trees from root to branch from bud to leaf.

Quite generally mathematical proof in the west changed after the introduction of calculus.

The main contributions to this were an introduction of fractions particularly continuous fractions, the argument by Berkely about Newton's Fluxions, the development of the number line concept and the notions of Dedekind, the formalism of the notion of function in russian and european mathematics, Boolean Algebra as a symbolic reasoning tool connecting philisophical argumentation (ie Logic) with a mathematical equation form; Cantors set theoretic work revolutionising the categorisation and taxonomy of just about everything, but particularly the abstract entities of line points and curves of geometry and number in arithmetic.

The Taxonomic movement of the late 17th century onwards lead to a hope of categorising and consolidating everything in neat boxes, which highlighted the consistency of a subject and introduced boundaries of categorisation. This was felt to be epistemologically helpful and academically a justifiable thing to do. Many research grants flowed from this effort and in a very real way it mirrored the Islamic effort centuries earlier to coordinate all knowledge in the empire, which lead to the significant mathematical categorisation of geometry algebra and arithmetic, and in the sciences Alchemy which is the foundation of chemistry.

It was Bertrand Russel who was the principal catalyst for the change in the notion of proof, as a focal point for many mathematicians and philosophers who were calling for greater rigour greater certainty in the foundational notions and methods and theorems of mathematics. David Hilbert is no less a champion for a more rigorous and defensible form of proof in mathematics. His aim was to vindicate the long held belief that mathematics was the only eternal verity among the knowledge of man besides religious formulations. Russel's aim was  to firmly embed mathematics in philosophical logic, as a branch of philosophy.

He embarked on this with his collaborator A. N. Whitehead with great vigour and dedication. using the most exacting tools of logic which we hardly ever use even today.He was inspired by Boolean algebra and boolean logic.

Despite the fact that he failed what he did was to change the course and mindset of academic  mathematicians to this day. He was the Father of increasing abstraction in mathematics, especially notation.

The Hilbertian school of thought managed to reveral that mathematics can never be proved to be consistent of itself, it needs to be embedded within a larger system to prove its consistency within that. However the same thing applies to the larger system and so on. The fractal nature of consistency and proof was thus revealed and the futility of Russel's main goal Highlighted

Mathematics benefits from rigour in its demonstration, but the abstraction cripples it even as it seemingly unites it.

Today it is more reasonable to ally mathematics with computer science, but the real font of mathematics is The Logos Response. This gives risse to both measurement and Language to distinguish the measurements and the framework to perceive and explore a geometry through proprioception, which underlies the manipulations and associations and syntax and procedures of spaeaking , describing and thinking and thus logic. Logic arises out of the proprioceptive geometry of The Logos Response as does mathematics in general, but measurement precedes these as does computational processing.

So one "saw" based on logic 101 made the claim that mathematics being based on all statement "A = A"   was inherently trivial.

Of course this is one falsehood placed on a truism, and is not trivial to explain.

A =A is the formalism of mathematics, but in fact it is not exclusive to maths. It is used every day in criminal justice systems in the sciences apart from calculations, in virtually every description we make using language. It starts off in general as B = A but finishes with A = A. These are not only procedural steps , they are state or stage notation and evaluation of a process, be it an argument, or an investigation, or a journalistic description.
Although this is an equality. it is used to denote an equivalence relation until the equality is accepted as an identity. Identity involves similarity and congruency, and even exactness, which are all subjectively determined by inspection procedures of one form or another.

As usual communication and notation are all subject to personal interpretation, but the marks are out there for personal inspection and by making the same mark using the same notation some measure of coherency may be established. Nevertheless these symbols have more than one use and meaning even in the most rigorous mathematical demonstration. This is why Russel was at pains to demonstrate that his system of notation was necessary to establish even the most "trivial" of definitions 1+1=2.

He showed that maintaining the exact meaning of a notation necessitated a circuitous method of proof, but he justified the ellipsis and subtle shifts of meaning in rigorous formalism as not being detrimental to the conclusion, and therefore rigorous formalism was a safe way to produce certain and logically sound answers.

That being said mathematicians use the formalism as a way of notating a discovery arrived at by different means and in a different order frequently. The procedural formalism thus becomes a test of the veracity or logical consistency of the relations in the notion, and in this sense they test or "prove" in the old fashioned sense the relations and statements with these relations being put forward.

Now for the reader they hold a different function: they are a meditative statement identifying a stage and a stopping point for meditation and reflection. The statement starts as B = A and B begs the Question which is answered by A, This is why it is called the answer. However this is not the full answer because the relationships may not thave been fully demonstrated. Therefore the next = means i have a further answer to the preceding questions. This answer is usually some transformation of form or manipulation of form that upon meditation and reflection reveal some other relation. And thus the presentation proceeds from equality to equality, from equivalence to equivalence.

Occasionally the meditation is halted while some related notion is reminded and brought into play, and then the presentation precedes to its conclusion.

Now this meditation is formalised in statements A = A, but the action involves manipulation of symbols or thoughts or geometrical objects or real physical objects, the inspection of relationships or measurement of quantites, the perception of forms, and relations and contexts and links and equivalences, and the motions of wholes or parts relative to frames of reference. The imagination is excited and the geometry is perceived in a new or more fundamental way.

Of course abstraction hinders this by making the referencing tortuous and in some cases tautological. I hope that by giving maths a new name Manipume‘ and building it from the ground up again that this fun, meditative and playful pastime might recommend itself to more ordinary mathematicians in the real world.

Hairy balls  (http://en.wikipedia.org/wiki/Hairy_ball_theorem) and cowlicks (http://en.wikipedia.org/wiki/Cowlick)!   sounds a bit saucy  :-* i know, but these i think are indicators of the fundamental reference frame of reality, certainly my experiential continuum. Along with the advice from my friends the spiders i have decided that the reference frame i have been seeking is not spiral but radial.

Thus the reference frame has a large set of radials radiating out from any "point" and these radials are orientations from any chosen radial. A particular set of radials is chosen as the special case and these radials ar chosen as straight lines intersecting at a unique region, which i will later define as a point.

Although radials are usually thought of in this way in general a radial set may be any set of curves that intersect in a unique "point" with no intersection at any other "point". This allows each radial to be part of a unique reference for another point.

The limiting set for such radials will be a set of trochoids without the loops, so from a straight line to the cardioid.

I feel like a gambler because these curves are now considered as roulettes (http://en.wikipedia.org/wiki/Roulette_(curve))

So maybe this is my lucky day and i have spiralled onto the right track! I am betting on Red  ;D .

Sexual innuendo and gambling! Whatever next! :embarrass:

Proprioception experiment 3
Warning: mind your fingers!

Its breakfast. so placing it before you shut your eyes. Now carefully find the breakfast and implements and start to feed your mouth,
Notice how you use other exterioceptors to locate where the food is relative to you and your mouth. Notice the visual map of your near surroundings and the additional details invoked by the other exterioceptors. It is almost like "seeing", but this is a generated image and map based on your other sensors and illustrates the density of the mesh of sensors involved in the internal re presentation.

The proprioceptive sensory mesh input can be distinguished by the geometrical grounding of the other sensory information. The kinesthetic whole links the proprioceptors to the exterioceptor input and is then represented as an output in all sensory systems visual, auditory, gustatory and kinesthetic( specific feel and touch systems within the overall kinesthetic).

Because of this kinesthetic coordination of proprioceptive signals and the other sensory signals, dynamic referencing is possible by using one or more of the sensory systems to hold or store a computed output. Thus i can compute an orientation in terms of sensory system output and turn to a new orientation while the process computes the changes necessary in the output system to maintain that orientation. Thus my muscle tension referencing the initial orientation "rotates" and twists around my proprioceptive mesh as i turn. However many of these tension sets i can compute and hold is how sophisticated  can be at spaciometrically analysing my position and orientation in the place around me.

Typically these tension sets are easily disturb and so my orientation is easily confused, but practice makes this skill more reliable.

Hence feeding with your eyes shut  is usually quite safe, but over a prolonged period i find i go out of synch and unless i keep refreshing the map i can go way off course. Hence the warning .

When you open your eyes the kinesthetic map becomes dominated and washed over by the visual signal input. The dominant signal input is isual because of the amount of processing required to produce a visual output, so the other signals interfere with a more dominant or greater amplitude signal and produce a carrier wave effect on that signal.

However each individual has a choice represented by the filters hardwired into the neurology and the selection by preference, habit, or design of the set and arrangement of those filters. These filters emphasise for each individual the main or preferred learning and referencing style. Thus a person may prefer to attend to the weaker but for them more significant proprioceptive signal or the auditory signal etc. Thus attention represents and consists in a choice of signal filters and a style of using those filters and an intention to use those filters.

Spaciometry makes no sense therefore without this set of fundamentals and the intention of the Logos Response : to measure, compare and distinguish.

In case one is thinking of measuring in the logos response as an active agent, this is not the case. Measurement is inate within the system of sensors: that is the purpose of a sensor i to measure and output its measurement as a graduated signal. The sensor mesh itself compares  and passes on a comparison as a distinction. Thus the fact that at the level of a conscious organism i do these things is of no surprise because this is the ineluctable structural function of the sensory system.

In my reality spirals and vortices are about rolling and twisting and curving and bending all of which are dynamic spaciometric rotations.

Rotations of what? Well space essentially and what inheres space. Thus carpets and scrolls and rolls and ropes and twists and coils and weaves and knits and knots and motions and matter whatever phase state and fields of motion whatever type.

It is interesting to think of fields of motion such as electromagnetohydrodynamical ones to gravitionucleardynamic ones as rolling and twisting and curving and bending. Many QCD and classical attributed properties of space which are apparently problematical will be explainable in a dynamic explanation of these fields as opposed to an axiomatic stationary one.

Found what i wanted 3d trochoids (http://trochoid.blogspot.com/search?updated-min=2010-01-01T00%3A00%3A00-08%3A00&updated-max=2011-01-01T00%3A00%3A00-08%3A00&max-results=2)! Roulettes have not got this far yet.

I woke up this morning and started thinking about matter and its cognate mass, related through the attribute of density to space. Ended up thinking that certainly in the west if we want a theory of everything we will need to veer away from 2 unacknowledged principles: space is nothingness and space consequently is static. Similarly any notions of the void in eastern culture must have the following two principles: the void is somethingness and that somethingness is dynamic.

With these principles i can see that trochoids and more recently roulettes provide a spaciometric topology of the motion field or dynamic somethingness of space that account for both classical and quantum Physics.

Usually one then says i have not done the math yet to demonstrate this, however i want to point out that the math only corroborates if you believe and understand the math, and the math in this case is topology or rather spaciometry. Anybody who can think geometrically can corroborate it for themselves. Whether i or they can convince others is not my concern. To each their own i say.

Vector's post here on this page
http://www.fractalforums.com/3d-fractal-generation/truerer-true-3d-mandelbrot-fractal-(search-for-the-holy-grail-continues)/60/

may be very significant in this regard. In any case it is illuminating . I found it so then and even more so now.

All Vector's posts are found here
http://www.fractalforums.com/index.php?action=profile;u=1056;sa=showPosts

And i hope he will let me know if he minds but i have taken the liberty of uploading his work into my gallery.

"When i was a child i thought like a child and did childish things....  " this quote from st. Paul is significant in that we all have to embrace change and revolution in our thinking as we mature.

 buckyballs (http://www.sciencedaily.com/releases/2010/10/101027145218.htm) and Buckminster Fuller (http://en.wikipedia.org/wiki/Buckminster_Fuller) a real example of geodesic curvature (http://en.wikipedia.org/wiki/Geodesic_curvature) see the glossary (http://en.wikipedia.org/wiki/Geodesic_metric_space#G).

Notice that a parameter is a choice of measurement technique, a way of turning geometry into numerals to utilize the properties of a set of geometrical spaces homomorphic to the set of numerals.

What we do is we standardise a geometry, we standardise a measurement fractal on a part of that geometry. we develop the algebra of the fractal on that geometry and then we apply the algebra to what seems to us as similar cases . When we do this the standardised fractal is called a parameter, because it is used alongside the geometry we are investigating. But it goes deeper. We actually dimension the geometry we are looking at using the standardised geometry so that we can use he parameter.

The dimension is how we cut up or boundarise the geometry and forms in the geometry. The parameter is how we measure the cut up pieces. So i can cut up 3d space into extended strips or areas  or volumes, and i can parameterise these by using a very thin strip as an extension measure; fractalise the strip for greater paramtric distinctions and dimension area and volume by radiating orthogonal strips of this fractal parameter. Each of these cuts (dimensions) is then measured by the parameter.

Now suppose the extended strip is chosen as a "straight" strip, we get a cartesian dimension of space, but if it is chosen as a curve we get a Riemannian dimension of space. Further if the curve is a closed loop then we get a "polar" or "spherical " dimensioning of space and the Lie Algebras that go with it .

We are more familiar with a mixture( straight and circular) of dimensioning curves when we deal with spherical coordinates, but this is not the only case. And hitherto i have been seeking a spiral dimensioning curve, but now realise that a trochoid set is probably the only uniquely parameterizable set of curves, at least up to the cardioid.

The Logos Response is a subject of intense investigation (http://www.sciencedaily.com/releases/2010/10/101026172021.htm) by scientists.  and the cold (http://www.sciencedaily.com/releases/2010/10/101027133146.htm) and the heat (http://www.sciencedaily.com/releases/2010/10/101027133144.htm) affect everything.

Enzymes as batteries (http://www.sciencedaily.com/releases/2010/05/100525094906.htm)

 The eye (http://www.newscientist.com/article/mg20827845.000-blue-light-taps-directly-into-your-emotions.html)  of the beholder.

Feeling blue? Do not listen to the blues!! (http://www.pnas.org/content/early/2010/10/14/1010180107).

Of mice (http://www.nature.com/neuro/journal/v13/n9/full/nn.2617.html) and men (http://stm.sciencemag.org/content/2/31/31ra33.abstract)

The Logos Response clearly has an emotional contingent as well, which makes blue skies very attractive and orienting.

Just a note about some spaciometric issues.

In general spaciometry deals with regions and boundaries of regions. Regions may be forms and boundaries may be surfaces.

The issue of a boundary is where or what is it?  I have repeatedly stated that it is a computed entity, and therefore iterative, which in my book makes any boundary a product of iteration and therefore a fractal.My  issue is the intersection of boundaries.

In spaciometry i would propose that a boundary crosses another spatially, in which case it is like to sticks crossing, or metaphorically, in which case the so called boundaries are actual regions that originate a a common source region.

Now the common source region is naturally a candidate for the term origin, and personally if the region can be bounded within a decreasing spiral that tend to the "centre" of the region, so that at each whorl, or properly trochoid, the origin of the two or more boundaries is evidenced i would be prepared to call that, under those conditions a point.

It is not so much that the limit exists but that i can arbitrarily define an approximation good enough for the purposes at hand. Being able to do this i would suggest is the intention behind the word "fractal" and encompasses for this notion of point the attribute of scale invariance and self similarity.

Although i have considered these points as origin, the limit process relies on them also being "sinks", that is a region from which or to which the boundaries are , in motion terms, moving.The spiral motion bounding such a region is crucial to the notion of the point originationg or capturing all so called boundaries if closed, or radials if not.

In a similar way the spiral or vortex motion is crucial to defining an axis, and therefore a notion of a straight boundary or radial.

I note as i write that i am preferring to use radial for any line originating from a point or sinking into a point.

I also note that the crossing of radials can only be approximated by drawing pencil lines and assuming that these lines are in fact origins or sinks, otherwise we are approximating crossing sticks or any 3d object crossing, and a thickness attribute is necessary.
Similarly if i roll up a boundary or radial around its sink or origin a thickness attribute is necessary. Therefore even in the point notion above the thickness of the origin has to be adjusted by negative magnification to apply the "rule" and that is why self similarity is inherent.

In spaciometry therefore scale magnification is also adjusted in limit processes, and is a determiner of when the limit process has a natural end.

Descartes (http://books.google.com/books?id=kAcncYRmGv0C&pg=PA102&lpg=PA102&dq=Descartes+on+laws+of+motion+leibniz&source=bl&ots=sIpFJ3D_zr&sig=9wk17BBpZCFdRI25fceOWauRWak&hl=en&ei=G4zRTKqZJNy4jAfE1ISpDA&sa=X&oi=book_result&ct=result&resnum=2&ved=0CBYQ6AEwAQ#v=onepage&q=Descartes%20on%20laws%20of%20motion%20leibniz&f=false) laws of motion compared with Newton's, but rather an example of a long line of evolving thought on motion and motion fields.

Laws of conservation therefore have there philosophical basis in a mystical notion of the immutability of deity. Laws of conservation have proven extremely useful, and that is their main ground for retention, i.e. they seem to work. However they have no intrinsic validity if they are not derived from an immutable "something". and immutability is hard to find nowadays.

It is also of interest to me that Descartes proposed that all other motions could be reduced to his conception of motion, and we find this simplification in the reducing of motion to straight line events today. Kepler it is mentioned thought otherwise, and accepted motion as intrinsically curvaceous. The consequence of this is to upset the Newtonian model slightly and to precursor the Einsteinian model greatly.

Newton and Descartes  (http://books.google.com/books?id=kAcncYRmGv0C&pg=PA102&lpg=PA102&dq=Descartes+on+laws+of+motion+leibniz&source=bl&ots=sIpFJ3D_zr&sig=9wk17BBpZCFdRI25fceOWauRWak&hl=en&ei=G4zRTKqZJNy4jAfE1ISpDA&sa=X&oi=book_result&ct=result&resnum=2&ved=0CBYQ6AEwAQ#v=onepage&q=Descartes%20on%20laws%20of%20motion%20leibniz&f=false) on  Force and inertia is an interesting insight as is Newton on Matter.

Newton's notion of matter is the more confusing and unnatural. In my opinion also unnecessary as Density is a separate attribute of mass and could have been clearly defined as such at that time. However Given that Descartes had no conception of the role of density in force and motion transfer it has to be seen as a step in the right direction.

Newton's notion of force is new. Descartes notion goes back to the greeks and though more obvious it is not based on observation in detail, or analysis of Galileo. Newton's notion is more mysterious: Force only appears in the action and application of it, and acceleration is proportional to the applied force, and inheres in he applied force. Plus force resists force initially!

What Newton was describing was Equilibrium both static and dynamic. Because it was not understood as an accelerator until Newton, Equilibrium and stability took some time to be appreciated.

It is important to realise that along with a motion field there is a concommitant equilibrium field, called an inertial field.This is the viscosity of space and the density of the region in which we have our being, and is a relativistic phenomena.

The motion field which we exist in has not been identified yet in the same terms as other field theories.
It will have to account for inertia, friction, acceleration, equilibrium, both static and dynamic and complex motion of "bodies" in various phase states. The behaviour of motion should encompass both wave and particle motion and spin at all scales.

When considering the notion of motion one has to acknowledge a rational framework for it and for Descartes this was a god, as it was for Newton and particularly so for Leibniz. The idea was therefore to correctly describe the workings of this god in the substantive realm, there being , as assumed by all , a non substantive realm. Descartes by minimal observation and maximal deduction from religious principles derived laws for the substance and motion in this substantial realm, but in ignoring observation committed "fallacies" to which Leibniz drew attention, and in particular created a problem with the insubstantial interacting with the substantial. Leibniz attempted to correct for this by invoking a kind of life force (http://scienceworld.wolfram.com/biography/Leibniz.html) in motile bodies, derived from god and not intrinsically related to matter in the cartesian way.

Newton felt inspired to see this force as being intimately involved with motion and proportional not to the mass but to the acceleration, thus extricating it from Descartes  matter notions and Leibniz unclear proportional relationship, based on Huygens and his desire for immutability to support his God cause.

Newton was Equally motivated by divine inspiration to set out accurately what his God knew to be the working of his motion field of matter, but this time based on acute observation. Thus the mysteries of the acceleration and inertia revealed themselves to his gaze  but through a 17th century prism that made force an interference from god or beings uknown who nevertheless obeyed god given laws.

Force was not defined because it was axiomatic that it was a "divine" push, transferred and maintained by bodily interaction.
Newton demonstrated that this was through change of acceleration,and noted resistance to acceleration in his laws of motion as inhering force. This subtle difference enhanced the role of density in his thinking, and the maleability of mass within a magnitude.

For Newton a body was a mass within a magnitude of space,but he could not seem to clarify this thought and therefore treated of point masses.

From Descartes to all following (http://docs.google.com/viewer?a=v&q=cache:pzqHVJdGPwIJ:homepage.mac.com/hermetic1/2245pt2.pdf+Descartes+motion+Leibniz&hl=en&pid=bl&srcid=ADGEESgjtwy80-T6SbXjMC3Icua9zffQDoio2XkS5umGstDIm3O2vZGYN2Y2vdBrjh4L4db7RZHOPK1xyS_2y-s0fzt3SjbEUGqUP60MC9U8_4EiUj2plPjQdzRO0Da_0U4UOl2VYm-O&sig=AHIEtbQ0hH_3aOhKolTi4RU9mkZ4oKxCsg) the substantial nature of space was dual : it contained matter and nothingness: God was in an entirely different untouchable realm. Thus matter devoid of its cause and in a non impinging space called nothingness is the root of the difficulty in comprehending motion and our motion field in particular

Spinoza attemped to make motion inherent in space as an infinite attribute. While Spinoza can be a difficult read because of the subtlety of the ideas and the comparison with Descartes overwhelming the contrast, it is clear on reflection that the Cartesian Axiom of Extension and motion being an inherent unity is what causes the differences in opinions.

Thus, correcting an earlier statement i posted Descartes and Spinoza And Leibniz delineate a chain of thinking starting with a substantive whole differentiated solely by motion to a dualistic whole which is intrinsically differentiated requiring insubstantial attributes such as god derived forces and immediate infinite action to maintain observed motion behaviour. Newton in his philosphy adopted a truely absolute spatial "nothingness" as a non impinging backdrop to his discussions, and thus firmly fixed the duality in modern scientific thought, by default!

I cannot think that Newton intended to decide the issue through his philosophy, but rather to provide the most accurate information and description of how motion actually worked that he could, admittedly as a divine revelation, but not as a declaration of war on Cartesian thinking.In fact Newtonians and Cartesians jointly attacked Leibniz school of thought, not each other. The popularity of Newton's mathematical mechanics and dynamics meant his working axioms were absorbed by osmosis.

As Newton did not set out to Debunk Descartes philosophy, there is no clear working out of how his working axioms, assumptions affect the Cartesian Economy, and indeed Einstein and others have paid little attention to the effects of the philosophical underpinnings, regarding the metaphysics as beyond them, and dismissing it with a few pithy sound bite comments.

The philosophy of relativity (http://www.amazon.com/Reign-Relativity-Philosophy-Physics-1915-1925/dp/0195177177) was overwhelmed in the existential (http://en.wikipedia.org/wiki/Existentialism) philosophy of the early 20th century, firmly placing the Cartesian Description on the back burners of history.

The relevance of Descartes is the unwinding of confused modalities in modern metaphysics of Physics for example; and for illuminating the notions we use everyday confusedly, despite having hard referents for them.

Spinoza by making motion immediate and infinite in his philosophy lifted the causal bond Descartes places in his philosophy between gods and motion, but without any causal replacement he has to make it infinite and eternal, ie he did  not think it necessary to have a impinging non substantial cause forever at work, for his own reasons, just as Descartes had his reasons for deducing such an impingement.

Leibniz for his own reasons wanted an inherent impingement, as an internal and eternal property of matter attempting to deal with the insubstantial /substantial problem of cause. Thus the problem derived from the insubstantial realm and its action of impingement on a substantial realm. Newton sidestepped this problem by dealing with the two realms separately in his work, hoping to reveal the connection through his work rather than to assume it.

Thus Newton's description of force and inertia are in context the beginning of his contribution to how the insubstantial impinges on the substantial in observation. That his method has supported along with Darwinism a general decline in belief in the agency of gods i am sure would be of concern to him, but that is the nature of cultural reworking of important mores, any idea that gives grist to the agitator's mill is used to promote change. We call ir spin doctoring nowadays.

I find after engaging in this study that i am reworking Cartesian notions in the light of Einsteinian Relativity theory,and Feynman QCD theory. It is about time i think that somebody attempts this.

Hermann Weyl (http://en.wikipedia.org/wiki/Hermann_Weyl)  

Spacetime (http://en.wikipedia.org/wiki/Spacetime#Spacetime_in_special_relativity)

Philosophy of science. (http://en.wikipedia.org/wiki/Philosophy_of_science)

I have found Kant (http://en.wikipedia.org/wiki/Immanuel_Kant) to be the one who advanced the slow unravelling of Descartes conception up to Newton, by advancing Newton's model, without his Philosophy. Therefore Kant provided a way forward into 20th century philosophy and scientific thinking.

I find in Descartes the kind of thinker who typifies the Great philosophers of the ancient ancient world. Thus Descartes could be Chinese or Indian or Buddhist or Babylonian, Islamic or a magi from the far eastern regions of Europe, or of  the natives or indigenous peoples of many islands and coasts.

What Descartes philosophy reflects is the gathering of knowledge afforded by Empire, and the industrious aggregation and compilation and processing of knowledge by the Islamic Arabic empires with its seat of learning at Baghdad.

FGrom this perspective Descartes is possibly the last of this old school philosophy where perfection is the organizing principle and the absolute is the self justifying basis and goal of thought.

Although Chinese philosophy encapsulate the no absolutes dynamic nature of "reality", it does do perfectly! Therefore they avoid the stagnation and inconsistencies of perfection within their philosophical model but have no falsifiable limit to it. tus they are supreme analogical thinkers, with a system of philosophy that will encompass any change or new observation by rearrangement of the emphasis on the elements of the old and by attachment of referents to existing "yi".

This makes chinese philosophy moribund, confusing and taciturn, requiring years of study to master even but a portion of its applicability. It is a blunt instrument for analysis but a marvelous repository of knowledge and information and categorisation.

Reform of the chinese philosophy by Lai Zhide for example reveals how new scientific insights can be clothed in ancient symbols and practices, giving a false sense, but a very chinese ideal of respect for the wisdom of the elders. This leads to a slow plodding approach to innovation.

Descartes was the tipping point for a period of ceaseless innovation in the west. Because Descartes was absolutist, he was vulnerable to attack, and indeed Spinoza reformed Descartes conception with his own absolutist ideas, and so on.

However Galileo and the great Leonardo Da Vinci were observationalists, called empiricalists. They were hounded because they were not absolutists, therefore they were against god and deceived by the antagonist to god, a very zoroastrian notion.

Some how what the magisterium feared and tried to prevent happened and the seed of empiricalism grew and finally toppled the absolutist paradigm, and Newton and Leibniz were crucial to the unraveling of the absolutist philosophy of Descartes, although only Spinoza at the time proposed an infinite substantial property not requiring any agency as an axiomatic possibility.

Today we still have absolutists contending with empiricalists and all shades in between! We could eventually resolve this in the style of chinese philosophy, but to me that is to miss a trick and an opportunity.

Benoit Mandelbrot provides me with a Type or paradigm of geometry which it is possible to philosophise into a system that i hope can resolve these issues in a modern, fresh and innovative way. Moreover it provides a substantial rethinking of the roots of knowledge (epistemology) the fundamentals of motion( metaphysics) the aesthetic response and the ethics of animates.

The Essential idea of Mandelbrot's paradigm is Roughness and i will address this idea in what i think are its essential components: iteration and approximation and computation and manipulation and scale/magnification dependence (self similarity).

   the question is can empiricalism deliver a grand unified philosophy? Furthermore should it even strive to?

I think any unifying idea or ideal is not an empirical product, but a computational one arising out of a pattern seeking algorithm, which is perhaps typified by iterative function systems.

This area of pattern analysis is probably best suited for constructing an overstructure for empiricalism.

The question of striving i think is an over statement as i think the algorithm organises in this way whether we are conscious of it, attending to it or not.

Sensors and sensor meshes are the basis of The Logos Response. Therefore how they depend on the underlying relativistic motion field is of interest and of importance.

At the level of molecular biology research (http://www.sciencedaily.com/releases/2010/11/101102130931.htm) is being done that supports a electro mechanical reductionism of sensor function.

This means that amongst other thing we can look at cell groupings and structures in terms of electronic circuitry.

I would go further and view any molecular structure as a proto electric circuit.

I remember developing two characters, Thermo and Electro to dramatise what happens at this atomic level, so i can see Electro building circuits and Thermo trying to destroy them, but all he does is help them evolve!

I guess i can give them partners: Therma and Elektra! No i am not going to give them babies today! They have only just met!!!

So when i was positing a motion field as an axiom for the set FS, that was stated in terms of relativistic motion and relativistic motion transfer, to which i would now add relativistic equilibrium or relativistic motion equilibrium. This to me would be the basis of all mechanics and dynamics including QCD.

This may take a bit of thought and exploration to see how the latest work on phase change (http://www.sciencedaily.com/releases/2010/10/101004101437.htm) models my ideas but i feel that it is worth it. This is not to say my idea are cool, just to say that this is what i am talking about, willis! :-*

Yea Leibniz, yea Leibniz, go Leibniz (http://en.wikipedia.org/wiki/Gottfried_Leibniz#The_vis_viva)!

There is a a direct line from Huygens (http://en.wikipedia.org/wiki/Huyghens) through Leibniz to the modern notion of Energy. (http://en.wikipedia.org/wiki/Energy) .

Leibniz vis viva (http://en.wikipedia.org/wiki/Vis_viva) was influenced by Huygens formulation of mechanical sytems in which mv^2 featured significantly. In particular the derivation of centripetal motion kinematics reminiscent of Kepler's law (http://en.wikipedia.org/wiki/Kepler's_laws_of_planetary_motion), mv^2 was highlighted in closed systems suggesting that it was conserved. Leibniz took a logical step toward identifying this with  the inherent god given "force" that keeps things moving, in distinction to Descartes who deduced an impinging force from god always acting making things move but in no particular direction, and according to a law which kept mv constant. Newton while still upholding Descartes mv conservation empiricaaly found that god's force appeared in accelerating bodies and resisted initial impression to move then swung round and accelerated the body.(An equilibrium system).

So there were 3 notions or uses of the term force which have been retained to this day under different names.

Only Spinoza (http://en.wikipedia.org/wiki/Baruch_Spinoza) posited eternal motion unaided.A review of his philosophy (http://en.wikipedia.org/wiki/Philosophy_of_Spinoza)

Man the thinker
man the handler
The footler
the tailer
with every part of him
she thinks
With his penis he thinks
with her womb she thinks
with their heart
in his heart
she beats out every breathing moment
of their intertwined life
apprehending
comprehending
prehensile
in every taut
and tautological thought
with which he measures the world
that she manipulates
As toys and tools and do s and don'ts.
Holding on to notions taught
As if they lorded over all
Why not let them go
And cease the day
Why not?
Let us play..
for we discover more that way.

Albrecht DÚrer (http://www.google.com/search?hl=en&client=opera&rls=en&q=durer&aq=f&aqi=g10&aql=&oq=&gs_rfai=) an artist mathematician who so symbolises what people at Fractalforums are doing.

The incredible melancolia (http://www.google.com/imgres?imgurl=http://richardelwes.co.uk/wp-content/uploads/2009/02/durer_melancholia_i.jpg&imgrefurl=http://richardelwes.co.uk/2009/04/10/drer-rhinos-and-snowflakes/&h=2835&w=2196&sz=2106&tbnid=hHopncIlqmzErM:&tbnh=255&tbnw=198&prev=/images%3Fq%3Ddurer&zoom=1&q=durer&hl=en&usg=__KO-UjR7goJzJnfsIDH9A4CV13X8=&sa=X&ei=evbcTNnaBcGGhQf5y6CXDQ&ved=0CCEQ9QEwAQ)

I once loved a woman,
  a child i am told,
       i gave her my heart,but
She wanted my soul..
No, don't think twice cos it's alright!

Bob Dylan

I   love my wife of 22 years more than ever. Happy Anniversary Mate!

Just in passing a force IS a change in velocity, precisely, The notion of force that takes form after Descartes actually is confused, although of course Newton describes it clearly. I prefer pressure to force because it does not require me to think of some insubstantial person pushing or pulling, which is the essential but non empirical referent of force. The genernal force is like the wind or an expanding or moving gas, and by usisng Boyles Analysis i have a sufficient and robust notion of force related to pressure, and need only to observe that this is resusltant in a change in velocity. The directional nature of force is not to be simplified as Newton did but to be derived by quaternion vector addition over the surface of pressure.

a wolf spider in my wifes garden said to me,
"the world is as it is because of radials and trochoids.
in this world, such as it is, if you want a flat plane
join precisely 3 straight lines into a triangle".
and then through his neusis he spiraled away.
i was left wondering and in wonder...

So for me this is the final part of the puzzle about polynomials (http://www.und.edu/instruct/lgeller/complex.html).

The history of polynomials starts in proportions and equations from proportions. The application of proportions particularly to geometrical concerns naturally leads up to the cubics and intimates the quartics and beyond. But the difficulty that arose through this research was the development of generalisation. Generalisation is called algebra after the arabic meaning restoration (http://en.wikipedia.org/wiki/Algebra), restoring the greek and indian methods of calculating in general.

Along with arithmetic i can derive the notion of algorithm and the 2 ideas about sum up the dimensions of algebra: Rules and methods of calculation specifying what where when and how and what is acceptable as a result. Some time between the greeks and the Arabs influence number took precedence over proportion and measure.

So when the equations started to not have solutions that were numbers, mathematicians were slowly forced to creep up on measure and proportion again (http://en.wikipedia.org/wiki/Eudoxus_of_Cnidus). At the same time the chinese and Japanese were developing solutions to equations that used a table or matrix form. They were developing an algebra of matrices for linear equations.

Before any advancement to polynomials could be made the issue of negative surds had to be resolved. Negative numbers had been introduced in all forms of mathematics worldwide but negative surds had not received a serious acceptance until Bombelli. Once Bombelli had produced his "Algebra the major part of the arithmetic",  other mathematicians were able to synthesise a theory of polynomials, and to think in terms of generalised polynomials, and finally to begin to use the taxonomic name polynomial to describe the field of study. And as polynomials developed so did matrix theory and determinants as it already had in the far east.

Along with the theory of polynomials the development of algebra moved eventually in several directions:Calculus, Cartesian geometry, Advanced Algebra including extension to Quaternions and ring  and group theory, function theory,Tensors etc and through algorithms and symbolic algebras computational science and computers developed.

Despite this fascinating development from Bombelli's impetus, mathematicians still have not acceded to the vital role of geometry/ topology in the foundation of mathematics.

With my grandchildren i would start in kindegarten/nursery with Fractals. These would be solid and real entities as in fractals in nature, ephemeral and fleeting as in fractals in soundscapes and as thrilling and as exciting as fractals in rhythm and motion.

From this great experiential base including others i have not mentioned i would by degrees construct the fundamental notions of boundary  form and shape , surface and density and viscosity and pressure.

From these i would define spaciometry and study topology and set theory to distinguish by notation and language form how we proceed to construct an algebra. Having introduced the young mind to such experiences i would drill down into historical geometries and attributes and develop the notion of proportion and quantity and ratio.
I would then delineate fractals and their use in scales and measurement, and reiterate the fundamental nature of iteration. only then would i introduce the historical development of number and a critique of the cultural iteration of numbering or counting.

Space the final frontier of mathematics, or rather spaciometry!(both readings).

I have come to an appreciation of notFS that i did not start with. NotFS is perceivable as space, but not an arid Newtonian space but a dynamic form that everywhere impinges at different scales through relativistic motion, relativistic motion transfer and relativisric equilibrium. And where it impinges on me is precisely where i interact with it on all sensory levels and beyond those by scale to levels of interaction i call quantum and universal .

These mysterious levels of interaction are truely beyond my comprehension, forcing or more exactly energising me to accept them as essentially existing but unknowable, but excitingly and surprisingly apprehendable: something i can devote my feeble and mortal life to with the fullest of satisfaction.

That i see iteration at all levels is a symphony of the essential harmony between the processes that govern me and my form and the inherent processes within notFS. My response , the Logos Response, reveals to me the apriori informationinherent within notFs which  i uniquely both discover, uncover, and recover as well as create and transform cause to exist in a certain form and relativity and initiate into my consciousness  and thereby uniquely store and retain it as a transformation of notFS. These transformations i call the set FS, what i know and have modelled from interacting with notFS.

Therefore randomness and complexity are not attributes of notFS, but of the set FS. This is not to say notFS is not random and complex, but that i cannot know if it is, i have only reacted to it at certain stages in my life and development by calling it so.

The better reaction i feel is to think of it as containing all possibilities, and thereby to approach it positively and exuberantly, all the days of my vain and futile life!  :rotfl:
In my approach to notFS i intuitively utilise iteration(unavoidable) and iterations of signals convoluting them into a format  called by me measure. These measures proportion notFs and spatialise it giving perspective and ratio and pressure maps in all sensory systems, utilising signal interference and thus inhering a notion, an intuition of iterative pattern making on the basis of an equilibrium pattern of forms called sensor meshes. These evolved forms of sensor meshes represent a transformation from biological molecular conformation networks, responding to basic quantum electromagnetohydrodynamic motions in notFS.

Thus the logos response generates my spaciometry and i distinguish my spaciometry by language forms which provide me with the basic mathematics of my computationally constructed spaciometry. Mathematics is language development and language development is mathematics.

Therfore i sought to rechristen mathematics as manipumæ.

I guess we just do not remember. Few of us di recall the first experience of consciousness because it is in the womb. But the first thing we iteratively measure through our sensory mesh is the dynamic motion of pressure and sound. And then when we are born we are ovewhelmed with the dynamic motion of vision and taste and smell and gravitational pressure.

In some animates the sensory mesh has evolved to provide apriori motor and spatial ability through "instinct", which of course is a genetic neural programme or operating system enhancing following and copying behaviour through chemical and sensory stimulation. Even plants follow the sun!

So we use our sensory mesh instinctively in and iterative way to capture the dynamic nature of notFS. How we organise this flux of information is the proper study of computation, but mathematically,by manipue we use relativity based on radials and meshes and we sense equilibria int hese meshes by rotational, translational, orthogonal and proproceptive sensors.

It is probably a longshot, but i think there is something about Chladni (http://en.wikipedia.org/wiki/Chladni_patterns#Chladni_plates) patterns that remind me of Trochoids (http://en.wikipedia.org/wiki/Clapotis).

So far i have found that few people are properly studying [urlhttp://www.cymatics.co.uk/=]cymatics[/url]
but fluid dynamics and high energy plasma physics have already linked trochoids to standing wave phenomena at distinct energy levels. So i will watch the developments with interest.

One question i have asked myself recently is how is light refracted around an edge in a thin slit interferometer, and i suspect it has something to do with trochoidal motion in 3d when light is polarized.

@ Einstein.

How ironic i must say
if god turns out with dice to play
or that we shall come to see
 that god is us , that's you and me
 and that is not to say
 that we shall live forever and a day
 but that the world does come about
by how are processing turns things out!

As usual lots of things to research on the significnce of Hamilton. Time (http://en.wikipedia.org/wiki/Time) and of course Leibniz views on it with regard to motion, randomness (http://en.wikipedia.org/wiki/History_of_randomness) in which Benoit makes a welcome appearance, Quaternions (http://en.wikipedia.org/wiki/Quaternion) through which i hope to place in my mind Hamilton's mileu and how he presented his discovery in different fields of maths and physics, starting with his battle with cartesian coordinates (http://en.wikipedia.org/wiki/Cartesian_coordinates#Definitions) and the introduction of versors (http://en.wikipedia.org/wiki/Versor),and his application to generalized coordinates (http://en.wikipedia.org/wiki/Generalized_coordinates),

After that Hamilton is subsumed in the general Lagrangian mechanics and distinguished as a Extension field in a more general study of classical mechanics and lighting the way to quantum mechanics and Feynman path integral (http://en.wikipedia.org/wiki/Path_integral_formulation) mechanics. Hamiltonian mechanics (http://en.wikipedia.org/wiki/Hamiltonian_mechanics) can now be approached as a "worked over" reformulation of Hamiltons notations and inventions, generalising them into other fields with dry and self important names which totally disconnect the general reader!
We need to look at the Legendre transform (http://en.wikipedia.org/wiki/Legendre_transformation), but after that it becomes all greek to me!

An overview of vectors (http://en.wikipedia.org/wiki/Vector_(mathematics_and_physics)) is instructive given the history discussed before in an earlier post. It is interesting to note that Newton's contribution to vectors is little recognised, eing overshadowed by his Principea.
the impact of compound interest in the development of Calculus i have noted before, and hidden in that is Newton's contribution to vector application, through Descartes diferrentials and tangents.

i need also to look at the Laplacian of scalars (http://en.wikipedia.org/wiki/Laplacian) and vectors (http://en.wikipedia.org/wiki/Vector_Laplacian) and spherical harmonics (http://en.wikipedia.org/wiki/Spherical_harmonics) which i want to explore in relation to trochoids (http://en.wikipedia.org/w/index.php?title=Special%3ASearch&search=Trochoids&fulltext=1) in 3d.

It is rather beautiful to me that i have these thoughts pass through me , hurrying on their way to some genius who will fully decant them like some special wine. i have not the "time" to write of the rather wonderful things that come to me, like the conversation about the probability measure that lies within the Lasgrangian exposition, or the fundamental difference that degrees of freedom make to an exploration of notFS without the hindrance of so called dimensionality, all conversed as if Richard Feynman had taken upon himself to share his thoughts in his inimical and charismatic way.

The notion that the motion field is  a discontinuous function with outputs from minus to plus infinity, but including 0 which being an improbable case has a region of applicability, outside of which the probability increases but discontinuosly,

the rather wonderful notion that the trochoids are a fundamental group of functions which could be substituted into a Fourrier equation describing systems, or that a quaternion with all its coefficients being trochoids would produce a rather beautiful description of so called chaos and randomness, or that the general trochoid is not that of circles but rather those of irregular spirals...
or that the notion of god has a very low probability,but the existence of an absolute gud remains a possibility! You see nothing can be ruled out because we cannot rule everything in..

To see and hear these things as if in some deep absorbing conversation is truely wonderful, hopeful, and inspiring, because i am glad that i am not some deterministic computational machine with randomness slowly being squeezed out as we find out more and more, but rather a trochoidal maniac, spiralling interestingly throughout space out of control, but seemingly able to mitigate that sufficiently to Enjoy being alive!

Any dynamical system based on 3d trochoids of a circle are amazing, beautiful and moving, but they do not gow or collapse, they oscillate. For this reason i feel that the more general trochoid needs to be explored. In this case a dynamical system will have the possibility of infinite growth or infinite collapse, with a small region of oscillatory like stability.

However at this stage 3d trochoids of the circle with variable radii are good enough tu have a look.

What distinguishes animates? Certain unique abilities or forms. But in what way are they unique: binary,or discontinuous but ordered ranking, or continuous scaling of similarity?

So binary, either it is different or it is the same eg does any other animal cook by controlling fire? (http://answers.yahoo.com/question/index?qid=20091002200942AAyT8pl)
Now discontinuous but ranked, it is different by some measure or value that compares against a list or sequence of important values. Things re ranked according to the list, eg do other animates utilise fire?

Finally a continuous scaling o similarity, every thing thst is similar is distinguished by another measure that is scale which has a continuous property, which means i can find a scale value for every distinction i want to make eg do other animates utilise temperature variation?

So from that while warming my tea in the microwave it struck me how that act right there demonstrated that heat was the rapid movement of atoms relative to there molecular and aggregate bindings.

So as some put it the kinetic motion of molecules and atom is heat. Feynman described it as "wobbling". But of course we know that "particles" are regions of space that have spin, therefore it is not unreasonable to expect this wobbling to be trochoidal, and not harmonic oscillation.

The difference this makes is that at certain levels of heating the regions would break apart impulsively rather than elastically as one would expect from a smooth curve oscillation like a sine wave. In phase diagrams this would be saying that a solid could turn straight to a gas if trochoidal wobbles of the extreme kind dominate, and to liquid if sine like trochoids dominate.

Loxodromes are also curves on the surface of an object, usually a sphere but also on a cone where they take on the name vortex or conical helix more readily. So one way of thinking about the Trochoids of a vortex is to think about the trochoids of a loxodrome.

Loxodromes form a nested sequence of curves as one moves toward the centre of a sphere along  a radial that intersects a loxodrome. Imagine as the sphere roles along the path of a loxodrome how the trochoids form a nested set of twisting curves at an oblate angle kind of tracing out a 3d set of twisting cycloids,sinusoids and helixes.

The thing about a cone is that it always rolls around in a circle if it rolls on it conical face, thus the trochoids have one easy order but the loxodromic trochoids spiral in or out depending on the surface spiral(loxodrome). This kind of dynamic system is used everyday by British Rail in its wheel bogeys. Feynman relates an interesting tale on how this was an entrance question to joining the fraternity of physicists at his college.

Now on the subject of vortex rings (http://www.oceanlight.com/log/bubble-rings.html). I guess it has not escaped attention that a stable vortex ring is an ideal candidate for the science fiction idea of "force" fields. If we could generate them in a descending cone of expanding vortex rings each stable ring on top of another spinning anti to it, it may make a viable mosquito net? :dink:

Another fun place (http://userpages.monmouth.com/~chenrich/index.html) to play.

Many of us work on the premise that if something is old and venerated it is profound. But here ps a conjugate  idea, because it is profound it is new and up to date.

Grammar of course has a mathematical structure. Before i found the Logos Response i did think that maths derived from Language in this way,giving structure, syntax, order and parsing to it through the notation and algebraic rules for their use. Now i think they are in act aspects of the same thing. Therefore if an expression has a conjugate in a algebraic form there ought to be conjugate to language forms. 
 Thus x+y has a conjugate x-y, therefore subject + predicate (http://en.wikipedia.org/wiki/Predicate_(grammar)) should have a conjugate subject - predicate. So what might this  - predicate mean?

There is a big fault that primary teachers have fallen into with fractions i think. Hardly anyone understands proportions and proportioning as a reasoning structure for exploring geometry. Insteradof its preeminent place it is reduced to skulking in the backwaters of historical development of fractions. The Lesser has replaced the greater i am afraid, seriously damaging our engagement with mathematics. The moan that usually accompanies the mention of fractions is not to be taken lightly, for at that time millions of children's hopes of enjoying what up until then had been a wonderful subject are dashed beyond salvation it seems.
If there is one subject that can usefully be left to Advanced level it is Fractions, and then as a historical footnote in the development of the numberline concept.

It should be replaced with proportion and proportioning, thus allowing a creative mix of mathematics art and music and dance to be brought together into the maths curriculum. From this we may derive future Albrecht Duerers or even Da Vinci's, and make sense of the spaciometry of our world in all faculties of understanding.

Ladies and gentlemen....Sir William .. Rowan...Hamilton! (http://www.maths.tcd.ie/pub/HistMath/People/Hamilton/Rays/AccountSystemsOfRays.html)  Applause please!

So i awoke wondering why my education in vectors had been soooo lamentable?,Why Cartesian coordinates had so constrained the field of mathematics? Why Descartes had had such an influence on mathematics and science through it, after all it is really a simple reference frame, aand when he introduced it nobody was that bothered.

It slowly dawned that it was the personalities, the camps, the warring groups in the grand game, the nationalities: that in mathematics there was as Hamilton puts it a "mathesis" a way of doing mathematics that was doctrinaire nationalistic, patristic and encrusted with anachronism and tradition.

Because of this english maths suffered a loss for over a century due to the Leibniz Newton farago, important insights were overlooked in the case of Grassman because they were not part of the old boy network, and they discouraged outsiders..didah didah didah... Nothing new then.

Rodrigues and grassman what they would have given to have ccess to the web!

Hamilton is a great Irish figure in distinction to the italian, german and general European figures in our part of the history of maths. Because of the web mathematicians of all ages and abilities can get together and critique create and contribute without the old guard control!

So this new frontier in mathematics won't lead to chaos...because wikipedia has shown that effective democratic controls can be put in place, and minority or special interests can set up their own group, without the need for this centuries long hostility and browbeating.

I know that evolution means it is inevitably going to be involved somewhere along the line,but we invented gods to control this sort of thing, so we should use them!

So Hamilton almost singled out the area of Algebra as the new Messiah for mathematics and the sciences due to the success that had flowed from it due to Bombelli, Cardano,Euler Napier,Gauss, Riemann,Newton,Laagrange and Laplace,, on and on as the simple cartesian coordinate mathesis, method tied together all those in the game of maths,both assisting and frustrating mathematicians in what they wanted to think about, to measure and to manipulate.

Even today the myth of number is used to convey concepts that are not related except algebraically. The emergence of a dynamic applied geometry was masked by a clinging to the number mathesis,myth and method. It is amazing to look back and see how mathematicians and physicists struggled to establish a proportinate measure concept of space, whichGreek and earlier mathematicians had in their geometries, and which was common up to the time of cartesian coordinates and beyond. 

Newton did not have to have a vector algebra to deal with vector quantities, or dynamic situations, but he did have to invent a new mathesis for dealing with dynamic situations geometrically. It was based on Descartes, but dynamic not static, that is why he called it fluxions.

Didn't he half get into trouble for it! Berkely Lambasted him later. Fortunately for the shy,autistic Newton he was in a respected position because he was right and bright, and the plague had killed off a lot of other contenders! Still he delayed publishing until asked to by Hooke, because of the criticism and prsonal attack he would be subject to.

In those days it was no joke to be besmirched as Galois indicates. You defended your honour with your life! No reason to disturb the frogs and toads  then, by troubling the waters unecessarily!

Descartes methods included small differences called differentails later. Newton studied these extensively and through them and compound interest formulae found the binomial series. With this and the fact that differentials were used to algebraically study tangents through Proportions he was able to develop fluxions as a way of compounding tangents to give a curve solution to a dynamic system.

A differential is thus a "compounding sum" of tangential proportions and gives a curve. But along with tangents areas under curves were being studied again by small differences related to tangents. The small differences of these areas under the tangents could be Aggregated directly and they became integrals and were seen as and shown to be the inverse of the mathesis or method of "tangeation".

The difference between compounding and aggreation was not thought that significant, and yet it is  a systematic use of vector addition using the parallelogram rule. It was probably hidden by the infinitesimal numbers or fluxions as Newton called them! These were everywhere evident to Newton because he had developed the binomial series and could see them vanishing away in the limit in the series! But what is overlooked is Newton also regarded them as the result of the parallelogram rule, without which he would have quantities but no direction. Newton needed both quantity or magnitude as they distinguished it then,and direction to trace a curve path by tangential envelope.

So by fusing cartesian coordinat geometry and euclidean geometry with algebra of proportions Newton created fluxions for dynamic systems.Leibniz came to it later but for geometrical purposes and without the binomial series which he did learn from Newton.however he did publish both in differential tangential calculus and integral tangential calculus before Newton, giving no reference to their correspondence or collaboration. That was the basis of the dispute.

So sir William Hamilton was not in a glass bottle when he did his maths degree, and he read and corresponded widely.Whether Grassman and Rodrigues were known to him is  question, but in ant case no one claims that he stole their ideas, rather that all mathematicians were looking at how to tackle 3d dynamics. Newton for all his brilliance was constrained to 2d by rhe mathematics of his time, and in any case geometry dealt perfectly well with 3d.

So between Newton and Hamilton a generation of mathematicians enamoured with a more symbolic approach as opposed to a geometric one grew up almost disdaining geometry for its lack of algebraic rigour!

Hamilton puts it succintly, they sought a deeper truth than apparently plane and solid geometry could give, there being no advance in it for thousands of years! Plus no one could make their mark in geometry, that belonged to Euclid! Riemann was giving it a good go though.

It is very simple: Bombeeli showed that there was an algebra that was prior to arithmetic, something that had not been realised until he stated it in his treatise. In addition he had shown that it made sense of the "mene" the √-n . Because of him polynomials were generalised and gauss proved a general theorem using these "imaginaries " as Descartes called them. The imaginaries lead to an explosion in the interest and development and applicability of algebra, in which fiels Hamilton was greatly interested,especially as it applied to geometry and space. Because of their notation and their history Hamilton saw the calculus and the differential geometries as a powerful algebra applying to the real world the secret of imaginaries! But coordinates mucked things up!

Much as Cartesian coordinates werethe greatest unifying simplistic idea, they also constrained the imagination and thinking.

Hamilton studied optics because he hoped to understand how rays (or vectors) behaved in the world. Once he had done that he had the technical framework to be able to dismantle cartesian coordinates. Maybe he studies Moebius as well as Malus (http://books.google.com/books?id=9QWYJsXRz7wC&pg=PA227&lpg=PA227&dq=malus+theorem&source=bl&ots=RzCIqV1ZoI&sig=XxooTCswBXEDhZ4qOW5RAU2tA4U&hl=en&ei=fpnnTJP7IIOohAe0u-W8DA&sa=X&oi=book_result&ct=result&resnum=1&ved=0CBYQ6AEwADgU#v=onepage&q=malus%20theorem&f=false),but regardless he had a geometricl structure linked algebraically to positions and directions in space. from this he sought a deeper connection with the imaginaries which are involved with rotation and therefore reflection equivalents.

Hamilton's theory on the Algebraic Couples (http://www.maths.tcd.ie/pub/HistMath/People/Hamilton/Couples.html) prepared him to be able to do his task. It is to be noted that Hamilton was in a club of Algebraists who were linked to the sciences, along with his friend Graves.

That the imaginaries inspired algebraist is seen in Hamiltons reference tot the doctrine(mathesis) of imaginaries and graves seminal but rejected work.

So i woke to find that these things called vectors by hamilton, both the line and the coordinate system were conceived by him to promote the extraordinary efficacy of Algebra as a valid and useful part of mathematics over arithmetic, and a a gateway into a deeper understanding of the world around us.

Hamilton's dream has been realised (http://www.math.mcgill.ca/labute/courses/133f03/VectorHistory.html) but we are not party to it because some have made it their aim to promote themselves,and to keep us in the dark. Would that we all could see Hamilton's rays of light, and his notion of the vector as unencumberd by coorfinates,freeing the geometry to speak to us of the best form to represent it in.

And now we have computers we can perform vectors as naturally as putting knect pieces, or Zome parts together.

For me the natural matrix for mathematiccs is geometry. Everything proper and useful springs from geometry, within geometry is motion form, surface relationship, size of all sorts : tensors ,scalars matrices and vectors; magnification, scale and affine transformations,symmetry of all types of measure and measurement tools, and tools of construction with methods of construction including neusis and arrangements of all sorts including aggregates ,bundles ,structure groups and basic operations of addition ,joining grouping and subtraction and division and separating and disaggregation.

In our response to notFS we intuitively create the geometric  background,and the measurement imperative with tool design based on fractal patterns and behaviours. So why did Number become so important? i think that the historical importance of number comes from a different subject to mathematics and its matheses. I think that Numerology and gematria have been confused with mathematics and took on an inordinate influence due to astrological and religious beliefs or faith systems. Certainly the number theory part of gematria is important, but not more significant tna anything else and not as useful as measure and measurement.

it was said that when pythagoras found out that some measurement could not be given an exact magnitude that he was dismayed, but that Eudoxus saved the day by developing a theory of proportion. At that time numerology became a seperate and minor field in mathematics , but no one recognised it in Greece.

Measurement is such a natural given in geometry that it is no surprise that vectors were identified and utilised by some in greek thought at the time, although not called vectors. There does not seem to be any early concept of vector in China, although i feel polar coordinate were natural to the chinese through the taijitu in the i ching and astronomical methods.

So really no one saw the need to develop an algebra of vectors until Bombelli , who did not of course se them as vectors but as adjugates to numbers or measurements. Bombelli was an engineer and saw them as measurements plus an adjugate . The adjugate was neusis but of course he would not have realised that, he focussed rather on the form "mene" the √-n.

What Bombelli had found was magnitude and neusis: magnitude and a kind of adjustment of direction of the measuring tool.

Later moebius and others in defining a vector as a magnitude and a direction left out the neusis, the "leaning toward" by adjustment and so missed the rotational element of a vector as well as the direction . Of course Euler took Bombellis work and restored the rotation but not the magnitude or the adjustment to direction.

It was not until Argand that the direction of the "number" became significantly linked with the magnitude and the rotation. I do not yet know what Grassman did in his Ausdenslehre the study of extension, but the full notion of Bombelli is really only repeated in Hamiltons work, which includes all the motions that should be ascribed to a vector definition: a measurment that has variable direction through rotation and translation.

Thus a +ib takes its full meaning, as being the translation part of neusis and i the rotation part and the measurement is given by       √(a^2+b^2).

Bombelli by giving rules formed what later became the algebra of "complex numbers" but now it should be recognised as the algebra of 2d vectors, which Hamilton extended over the hyperreal measures so called.

C-C-C-C-C-C-C-C-C-C-C-C-C-COMBO BREAKAHHHHHH!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! O0

its about time somebody else posted in this thread. (google 'the bloody board')

By regarding gravity as a pressure field or system and looking at the ocean depths as an analogy the simple question why do things float, in particular why do gas bubbles that originate at the bottom of the ocean rise  relative to a gravitational pressure field, if such it is?

The observation that gravitational pressure causes a condensing motion is utilised as a starting point to think about this curious behaviour of a bubble rising when it originates at a place where the surrounding water pressure field is high but not uniform.

The motion indicates that the system is not in equilibrium despite the bubble presumably having a relative internal pressure equal to its surroundings.

Algebra I feel now has grown up or at least the cell division for growing up has taken place.

One cannot form algebra without analysis (http://en.wikipedia.org/wiki/Mathematical_analysis),so within every playful algebraist is a keen analyst and observer.analysts as you might imagine are quite a weird bunch of people even for mathematicians, so it is good to keep the quality I think grounded and not abstracted into isolation.

As usual analysis distinguishes itself from algebra and algebra from geometry and maths from physics etc... But really the relational links are greater than the relational disconnects because the same human workers founded the mathesis or doctrinal methods in all. It is the doctrinal  part that drives the separation.

I pretty much think I have a proto structure forming in my mind for a modern ontology of manipume based on spaciometry adult algebra!   and calculii of various sorts with arithmetic hopefully being an example of an early example of this taxonomical structure.

Algorithm. When I use the word I tend to use it in a more general sense as a specification of a set of motions that are sequential and iterative through a branch node specification which is based on the test operate test exit cybernetic principle. The algorithm is a kind of predicate to the entity it develops if you think of the entity as a sentence describing its motions.

This rather general gobbledy gook means that I can pretty much use algorithm in most transformation cases of interest and can make analogies with differing systems and within systems.

Analogous thinking is probably one of the first recorded types of thinking we human animates described, but of course it is a feature in all animates by degree. How very self similar and how fractal!

(http://www.wackerart.de/gallery/firewall.jpg)

I used gravity for the construction of some of my pictures.
http://www.wackerart.de/gallery/gravitation.html (http://www.wackerart.de/gallery/gravitation.html)
http://www.videocinema.de/film.htm (http://www.videocinema.de/film.htm)
Working with water colors can lead to a very fractalised picture.

Thanks James. Had not realized how intense it had become, while trying to work something out in different posts!

Just re read a bit and see typos and mis communications all over, so will try to tidy up. Hopefully got a render of a 3d hypotrochoid to post done in quasz, so that should break things up more.

Some of what I right may be right some wrong. That is not a problem to those with there eyes wide open, but I make the statement for anyone who may be tempted to think I have got it all right!

For example I have only just read a short biography of Descartes !! So that puts things in perspective. Descartes did think he was better than anybody else before him in describing reality, but that was just Descartes and I am sure he was an admirable fellow otherwise. We all have a delusion of grandeur anyway and we all certainly are hypocritical!

Thanks Herrmann, great illustration of the concept.

I have no hesitation in saying that Bombelli preceded Descartes in using an orthogonal coordinate system (http://www.encyclopedia.com/doc/1G2-2830900516.html).

"The linear representation of powers, the use of the unit segment, and the representation of a point by “orthogonal coordinates” are some of the noteworthy features of this part of the work"

This is not to diminish Descartes, but to draw attention to the fact that coordinates are not of his sole invention,and very likely he was influenced by Bombelli's Algebra, and that of the italian school.In any case triangulation for positining a point had been a long established geometrical application, so positioning a point even in 3d was not the importance of the coordinate system of Descartes.

Bombelli as i noted previously used a kind of set square tool to find roots of equations expressed algebraically by neusis. Although these were geometrical constructions it would not reveal the complex conjugate pairs without a representation in the plane of the ordinates of measure and sign.

Bombelli clearly did not have the influence of Descartes through influential friends and contacts, but the boy done good for his time and influenced algebra in the popular voice for nearly 400 years.

As with all these things nothing is ever final, and my research has shown that algebra and analysis were one and the same until the differential and integral calculus overwhelmed the majority of "lesser" algebraists and their analytical abilities. Intellectual differences/snobbery i think lead to those analytical algebraists who could do diff and int calculus differentiating themselves from those who wanted to analyse other things!. I also found the distinctions made between analytic and synthetic geometry.

Nevertheless analysis is still a subset of a wider algebra, no matter how good analysis thinks through its proponents it is.

Descartes in his co ordinate system, using them algebraically thought of the reference frame as being whatever it neede to be: static for fixed points,moving or at least having a linear velocity attributed to the parallels for moving points etc.

So the coordinate reference frame was itself an algebraic notation, which could denote measure or direction of motion or axis of rotation or point of rotation, or comparison of measures quantities etc,

Descartes influence was not mainly his coordinate system, but his method of algebraic analysis (http://ualr.edu/lasmoller/descartes.html)  and this is called a praxis , a mathesis or a doctrine of philosophy on Algebraic method. To Descartes his coordinate system were a tool of analysis ,for the carrying out of his method, and represented the simplest analysis of geometrical forms that he could conceive, and thus was the starting point of his form of analysis. From this start he would add in anything that naturally presented itself as being necessary in the analysis , but only after exhaustively manipulating the simpler analysis to see if it was sufficient to achieve his goal.

Thus Descartes praxis was governed by necessity and sufficency, and by this he hoped to arrive at solutions economically in terms of analysis, for as the Kohleth says" of the making of books there is no end!"

To this end Hamilton seemed to desire a similar influence on method through his quaternions, but this was at a time when Cartesian philosophy was a waning influence and the defense against the skeptic was anathema to the very progress of science.

What Hamilton achieved is all that he could in this time a recognition of his inspiration to mathematicians and scientists around the world for shaping the field of vectors in a useful way.

Manipume¨ i would say is devising methods of proportioning, whether by algebraic analysis or happy intuition, playfully arriving at some solution to some engaging problem, and thereby being delighted!

By analysis found out basic orbits are trochoids. Have not worked out an interesting 3d formula yet but this is a start.

I feel i have come to the end of this research, and by end i mean telos: that i have found what i was looking for even though i could not enunciate it; conclusion: that i have come to a conclusion or set of conclusisons which resonate with me and are of a fundamental explanatory nature to me; transformation: that i have come to that point, moment, place, space where transformation is occurring from who and what i was to what i am becoming.

For me there is no beginning or end as commonly put, just transformation between states, as commonly put.

I feel that i have fallen out into a space that is a dynamic magnitude,
A stillness that is not still and a quietness that is not quiet.
I know the expansion is balanced by a synchronous contraction
That a skew term is what is needed in our "transforms" as well as an "opposing rotations","opposing reflections" and "opposing magnifications" operators generalising the notions of translation and rotation, reflection and magnification so they can occur simultaneously and in opposing "directions". This is so that i can in general fold or tear a piece of paper, or curl up into a ball and go to sleep and explain it by a symbolism that accurately describes it.

I know that since Eudoxus mathematicians have known that the natural numbers are named proportions and are intrinsically scalars; that there is no real entity called a number, but that we have devised this as we have with all our tools and measures as fulfillment of our inate intuitive desires to measure the incomprehensible magnitude which we realise we are in and a part of.

I know that topology is the enclosing concept of all geometries, but i prefer spaciometry, and in our spaciometry we have struggled to piece together a proper appreciation of the magnitude, that it is not just proportional but dynamically proportional, and whatever else dynamic means it means motile in every measuring reference frame tool we care to construct to analyse it.

I know that the reference frames should not be used to delude ourselves that they are inherent within the magnitude, but rather they are tools we have created to explore and analyse the dynamic magnitude, which i will now call dynamic space( DS).

That our reference frames for DS are analytical measures which are not to be ordered or compared to numbers but to define kinds and styles of measure from scalar to tensor, including of course vector and matrices. These styles of measure have their own utilitarian function in our analysis and together begin to form fundamntal algebraic concepts of the spaciometry.

I know that it has been fun and that manipume¨ is a catchy tune!

Finally  :toast: i think that DS is aperiodically iterative, by whatever measure we may construct to measure iterativity and or periodicity, and therfor produces aperiodic fractals of which i am one.

That to measure this DS the spiral/vortex shell form is necessary with the spherical  shell being the "unity" of this topological group.

We will have to develop our topology of closed and open motile forms, developing suitable analytical measures for this fun thing to do and checking whether we can transform between the two in some algebra that makes sense of what we are looking at spaciometrically, always checking against the spaciometry for sanites sake!

For me trochoids must form some  irreducible component part of this mix, but the trochoids of the open form not the closed form, and the simplest examples of an open form to me are the conical and spherical vortices found in fluid dynamics.

Descartes by the way had a theory of vortices, but it is Hamilton who perhaps has lead the way to devising a suitable analytic measure which is directly spaciometric. I will have to look at clifford et al to see if any advance has been made in anlytic measures of this sort. Penrose's Twistors come to mind,but i am not familiar with them.

Of course i am not going to stop posting interesting tid bits  :embarrass:

Well i think they are of interest to me and i need to get the words out.

Eudoxus is of fundamental significance to all of western culture as is Theodorus. I know that the chinese nine chapter had a system of proportioning but i do not know much of it yet. Eudoxus howevever is the protypical weights and measures guy! He established the scales of measure and the theory of proportion and proportioning for greek society and western culture.

I do not yet know which cultures valued the void as a cosmogeny besides the Indian and the Egyptian, nor which revered unity as the cosmogeny besides the zoroastrian and the Egyptian Atun dynasty and the bhuddist philosophy school and of course Judaism which is an offshoot of zoroastrian ideology in one sense. I suspect the Greeks valued unity or at the very least the pythagorean school did.

So when Theodorus elegantly showed that there was no fundamental unity it threatened a whole lot in Pythagoras mind , because he portioned the magnitude of space as a multiplication of a fundamental unity. Thus the natural numbers have always been proportional, but in a scalar way as multiples of unity. This is why they were distinguished as Integers that is proprtions of unity,scalars of unity.

Eudoxus restored pythagorean equanimity, but they wanted to hide the information Theodorus had brought to their attention, i guess, so maybe he did not survive the social bomblast he caused. It was left to Archimedes to refer to his work and his proof of the non existence of a fundamental unity using the so called pythagorean theorem or fundamental relationship between integers.

What Theodorus showed was not understood or welcomed, but it was rediscovered and gradually embraced by western culture through Archimedes, who tamed its consequences by establishing the archimedian rule of magnitudes, a pragmatic approach to ratios and proportioning based on Eudoxux which essentially was: deal with what is necessary and sufficient when proportioning. So really it was the indian culture and its love of the void that found no harm in their being no fundamental unity,and allowed for infinitely large and infinitely small, with limit placed only by human perspicacity and endurance to iterate.

Why did Archimedes revive part of this social outcasts work? because of its obvious and beautiful linking of the circle, a respected "perfect " form to the spiral form. Archimedes had found a utilitarian function for the spiral and only Theodorus had a mathematikos on it until he started on the subject. It is to be noted that Archimedes eventually defined spirals in terms of a ratio of motions which bypassed the need to use the surd roots in its description and allowed unity to be used again to define even a spiral.

The spiral also provided a link between direction and measure and established the carpenters rule as the first "model" of a vector, and a great aid in neusis for Archimedes, as was the spiral. Archimedes using the spiral as a neusis path and a rule, likely to be a set square, found several ways to trisect an angle accurately. He also used it in the approximating of π.

Pi was allowed because the circle was believed to be a product from the gods and therefore magical. It was hoped that it would reveal its secret integer ratios to the devoted, instead as it turned out it ultimately revealed all so called numbers are baseless: they have no fundamental unity. It also revealed that numbers are scalars, that measurment is a vector action that trigonometric ratios are fundamental to any analysis of any measure of magnitude, and that they are scalars less than unity, therefore the basis for the notion of fractions; that the pythagorean theorem is a metric for all measures of space, and that right angled triangles are fundamental to any decomposition of the magnitudes of space. The circle also supports a ring group algebra of unity and sign, and vector bases,and the generalisation to the sphere does the same.

The unit circle has taken a pole position in Euclidean and noneuclidean geometries and is a fundamental analytical tool. It has also been used analogically to establish Algebras of non euclidean spaces and geometries and the development of dynamic vectors and algebras like complex algebras, quaternion algebras, clifford algebras, Lie algebras and musean algebras.

So where is the analytical tool for spirals?  i believe the sphere the cone and the radials have to be combined to form this type of tool, and that its fundamental components are the trochoids of a spiral which is simply the trochids of a dynamically radially expanding circle or sphere or alternatively the trochods of a  sphere projected onto a cone or of a circle onto a  line angle tangential to the  circle, or rather a rolling expanding circle between the line angle. This is not a clear description i know but only animation and exploration will make it clearer even to me.

As many have remarked Bombelli's wild idea has come up trumps.

Because of an inate pythagorean attitude scalars were promoted as the solution for all our geometrising. For a very long time even up to now the goal of geometry was to produce a scalar! this is why the √-1 was met with such derisison and incredulity. What √-1 reminds every mathematician and geometer that the goal of geometry is to devise methods of proportioning, that is algorithms  that proportion. What √-1 means is that within your method of proportioning you need to rotate!.

Of course we need to do more than rotate to proportion, so there are special terms that must relate to these motile actions i guess, Now i wonder what √-p does where p is any prime? Or even better still

Hamilton's Analysis  of the algebra of imaginaries for quaternions and Graves for octonions were important but ahead of their time. When i look at Hamiltons's analysis of quaternions i know that the quaternions drop out of a more general formulation. Thus Hamilton's Analysis has many more applications than just Quaternions. I am sure that more of the roots or -1 are covered by his system but he was constrained by the cartesian coordinates to apply what he had developed to the physics of his day. Hamilton struggled for the rest of his life to find applicability and meaning in his  system, and so did everybody else!

He found some powerful uses but not fast enough to dominate the analysisi that many physiscists were doing, and not in an easy predictive manner. In fact quaternions are now seen to be superior because they simplify the description of the geometry, just like Descartes system did, but descartes applied his system to the past not the future,and thus establised its merit well before it became an essential tool to analysis and the methododlogy of a generation of geometers and mathematicians.

Among the scalars of unity the primes (http://www.mlahanas.de/Greeks/Primes.htm) play an interesting part. They really show that scaling as an operation: that is proportioning and ratioing can be sorted into independent scaling "seeds". So for example if unity is fundamantal then the next fundamental scalar is 2. This means that i can reduce or scale down certan ratios to a "seed" of 2.
Three is the next fundamental scalar because of course bundling in 2" will not capture 3 and similarly i can scale down certain ratios to a seed of 3. It becomes apparrent that the prme numbers are interellated by scaling as 3 can be scaled to 6 as can 2. Systematically going through the scalars to find how scaling works, that is an analysis of scaling reveals these "seed" numbers which are all scalars of unity but which unlike unity scale to pick out only certain numbers.

To a geometer this evidence of a kind of mesh, ameshing together of these seed number scales explains the fundamental nature of the unity, and describes in a way how all thingsdeveloped from unity. Of course unity had to be male!

This mesh was related to an actual sieve mesh by Euclid, and a spatial arrangement was used to describe this curious mesh arrangement, by relating it to finding the area by multiplying the side lengths of rectangles. This mesh partitioned rectangles into equivalence classes, and related directly to proportioning  as well as to bundling or packing.

So for a while Pythagoras had a nice little thing going there with is theory of unity being fundamental, and worthy of study for that sake alone! Theodorus, shut up!!

What Theodorus showed was that primes or proto arithmetical objects, with the pythagorean theorem did not produce scalars of unity, or eve scalarsof proto aritmetical objects, so even the prot objects were not "proto" in that sense.

My guess is that Theodorus got the short end of the stick (http://www.mlahanas.de/Greeks/Irrational.htm) if not the sharp end of a dagger! That is how important it was. It took Eudoxus to restore equanimity and maintain"unity" by explaining the arithmoi as scalars the solution of all proportioning, and that these irrational numbers could be proportioned and scaled among themselves. Thus the arithmoi maintained their foundational and "creative" status status to this day. Some like to stand in awe of number in the style of Pythagoras even now.

So geometric measure and the right triangle "rule" with the aid of the arithmoi are seen to give birth to a whole new set of ratios,never before conceived, but children of tha arithmoi.

When negative numbers were introduced, as debt mainly,  their geometric nature demanded an origin, which is why the Indians were far advanced in their use and consequences and their sign rules and also their consequence of √-1.

For me this leads to the measure being the fundamental arithmoi: arithmetic and geometric object and as theodorus showed, the unit circle and sphere being the fundamental proportion or scalar, and consequently the mesh of prime measures being spiral in nature.

Of course if a circle or sphere is a proportion this makes an measure necessarily a vector, that is having magnitude and direction, and square rooting a geometrical operation or algorithm. In fact it reveals that all operations are geometrical operations not counting ones. Therefore, and it took a long while to realise this we need to define all operations geometrically and rigorusly.

Descartes began to do this in his Geometry and continued to do so all his geometrical career. Thus Arithmetic was sidelined as a practical case of a more general geometric construction, with a restricted set of operators. However mathematicians clung on to their familiar arithmoi with a religious fervour insisting that all True mathematikos should make them supreme, including their field properties.

Over time algebraic analysis and categorisation has lead to a downgrading of the integroi but has maintained a commensurate high regard for the field properties. However i think we have to find a geoemetrical definition of the field properties or behaviours if they are to survive as algebraic analytical fundamentals.

Thus Bombelli  using a set square vector with neusis has in a practical and applicable form all the elements so far alluded to in this story": the integral scales on both perpendicular/ orthogonal sides of the set square the incommensurable third side, the trigonometry, and the neusis, and the √-1.

From this geometrical start, polynomials of all sorts flow, and their inherent nature is that of a vector algebra of geometry. Polynomials are vectors and an algebra of vectors and a training in vector math that was of course not understood in this way, because "number" a translation of arithmoi was in many peoples head, along with the pythagorean and archimedian doctrine.

It was not until Descartes that the doctrine or matheis substantially changed, but the √-1 he could not conceive of. He did not have the "intelligence" of Bombelli, who in his travels may have gleaned that the indians had done work on this, and trusted to his instinct or intuition or intrguing discoveries and meditations. In any case Descartes had little time for it and derided it elegantly. His name for them stuck and in fact inspired an interest in them that lead Euler to find his remarkable equation and gauss to prove the fundamental theorem of polynomials, which i now regard as a fundamental theorem of vector algebra.

You have all heard of        

And of course many here know        

Which Euler arrived at because of his love of power series polynomials, which we would call infinite vectors today. Just as Einstein showed through polynomial transformations that the "rest" motion of a body is not zero but a scalar of value m*c*c

Euler showed that the infinite vector has a form directly related to the trig functions on the unit circle, which describe a continuous series of right angled triangle measures with hypotenuse of unity. These right angled triangle measures relate directly by symmetry to Bombelli's set square and is the reason why i define the set squares as vector models

Formally and fortunately convergent Eulers equation is the definition of unity for all planar vectors. it therefore must be a basis for all planar vectors.

I will have to check but i think this is what Hamilton established in his seminal work on couples.

Hamilton therefore looked for a generalisation to 3d and at the same time a change in the doctrine of numbers up until then, for he could see that the doctrine of imaginaries was a better or more useful mathesis for algebraic geometry than those of former times. He had a hard time but couples at least maintaned the field properties.

He gave up after ten years of looking for a solution in 3 variables, and looked at four variables. By this i mean he attempted to reduce infinite series to an algebraic system using the trig function series (Fourrier analysis) , the x polynomial series without success because he was seeking a scalr value for i*j to make sense in completeness terms. Thus when he eventually abandoned the attempt and moved to 4 variables he had substantially done al the work, and was able to show that an infinite series of quaternions converged to a quaternion.

So using a polynomial vector Hamilton established a general vector math after Cotes, Euler, Graves in the plane and another one After Euler with 4 variables  (http://mathforum.org/library/drmath/view/51639.html) which he took to represent a vector with a scalar time component.

Thus the 3d vector is given by  

"The quantity
(check it
for yourself).  And you can go back to the infinite series to see that
exp Lx = cos x + L sin x, whenever L^2 = -1 and x is real.  So let
, so  
Then
"

Hamilton realises that to make sense of these values they have to be applied to arc lengths on the surface of the unit sphere, thus establishing a use for a radian measure (http://www-history.mcs.st-and.ac.uk/Biographies/Cotes.html).

I have in fact turned up a new hero in the story, a certain Roger Cotes (http://www.mathpages.com/home/kmath192/kmath192.htm), who prior to Euler took great interest in Napier's logarithms and discovered
      



Which is every bit as wonderful as Euler's because it is one of the first uses of the radian measure that Cotes invented, linking the imaginary surd to the infinite iterative sequence of surds that napier worked with entirely by proportioning, and maybe usisng his napier rods as a geometrical aid. Thus it s doubly satisfying as even though Cotes used quadrature easily and so was at ease with calculating series of finite or infinite length, the result speaks entirely of geometry as fundamental to algebraic understanding.

We often hear the mantra that everthing is connected, everything is one!

As there is no fundamental basis to one it is probably more accurate to say everything is connected through everything being transcendental.

Hamilton immesiately refers to these as a system of equations (http://www.maths.tcd.ie/pub/HistMath/People/Hamilton/Quatern1/Quatern1.html), thus leading me to note that this 3d vector form is formally a matrix algebra on the 3 x n matrices. I do not know much about Banach Algebras and will look into it, but i suspect a link.

Hamilton spent most of his original presentation showing the constraints and equations necessary to transform from a spherical geometry to a cartesian. In this he reminds me much of the initial apects of tensor theory, which leads me to suspect a link with tensors which would become more obvious the higher the set of ntuples used.

I have to remark that Napiers logarithms (http://johnnapier.com/table_of_logarithms_004.htm) along with greek spherical geometry and trigonometry advanced through al Khwarzim is the basis for our notion of algorithm and its incessant iteration. Thus mathematicians of old were well aware of and embraced iteration, but it needed Mandelbrot to point out its geometrical implications outside of the mathematical disciplines. He really simply pointed mathematicians outward instead of inward. Like Bombelli he said: i can.., we can! And he looked at the bogymen of maths and said "you know what, they scale!, And i think you will find them rather beautiful geometrically."

Cartesian coordinates are not a fundamental measure, they are a tool of the cartesian method which invites the use of any other additional tool.

Spherical geometry is the natural measure for the quaternion system, and the measure system i would devise for that is : 1 radial pole for the origin. Then great circles as planar decomposites such that the radius of the great circle has the same magnitude as the spherical pole. Finally 1 radial rod for each great circle. The angle measure betwwen the rods and the pole are radians  

Using this as the construction basis i can define the constraint on any appropriate system of spherical measure to suit. This system of measure i will define as what i have envisaged radials to be

an observation
We can rewrite the transformations of the plane using matrix operators, and without quaternions we have to use matrix operators for the transformation of the 3d space.

I therefore would expect that moving to generalised coordinates would involve tensors.

I had not suspected, nor even was i taught that logarithms were based on the properies of the trigonometric scalars.

I had just started reading Napier when this point was first adduced by me in puzzlement and then deduced by careful reading of the text, and then boldly stated by Napier!

Therefore to me the mystery is revealed in Napiers invention and method! (http://history-computer.com/CalculatingTools/logarythms.html) and what at first seemed strange now seems inevitable and consequential!

By his invention Napier has given a logarithmic basis to all measures of spaciometry,and an alternative decription of the aggregation and disaggregation of bounded quantities/magnitudes. The scalar mesh lso becomes amaen`ble to logarithmic description as do all vectors, matrices and tensors and beyond,

Logarithmic operators must exist and may form an Abelian Group.

I see also the reasoning behind Hamiltons couples as an extension of Napier's explanation of the development of the logarithm of sines!

I may have overstated it but it is a fascinating find to me and explains why the term logarithms is used in distinction to powers in polynomials, a question i have had since a child.

Indeed all things come to those who wait!

"  they also serve,
who stand and wait..."

Another rather startling insight From Hamilton's work is the notion of vector.

Most simple definitions link magnitude with direction/orientation. Magnitude is an old fashioned word, but mathematicians still use it for a procees of squaring and taking the positive square root of a number or quantity. Quantity is a more recent word but does not have a mathematical algorithm attached. and denotes a magnitude or amount of something. Amount is another word used to describe the same thing.

Mgnitude is actually linked to the word magnify,and then we have size and bigness and even mass which has a physical significance attached. So these terms sit uneasily nestled in the mind with the words matter and space. 

I generally use an activity to define these ntions precisely for me. The activity is extension, a proprioceptive action of extending or stretching or reaching with any suitable part of my body.

Extending is what proprioceptivly through kinesthesia gives me the sense of "extension" called in olden times "magnitude". The fisherman's tale is a fable of magnitude that is extension demonstrated by stretching out the arms. Extnsion, therefore ,is the notion of magnitude.

But extension is always directional! so our basic notion is and always has been vectorial!

We multiplex these vectorial sensations to apply to a volume of space. A volume measure then is always spaciometrically rotational built up from multiplexing vectorial senstions.This is our notion of the magnitude of space also called in general magnitude.

The shapes and forms hold for us a memory of these magnitude sensations by projection/perception-recognition. This really is the limit of our sense of magnitude, the rest we progress by imagination, which means we bring the outside into our model of sensation and adopt an observer view proprioceptively. This is possible because the mosel is a vector entity based on active vector measures, This type of model we have begun to call tensors.

So magnitude has always been this tensorial vector matrix or mesh. How do we dimension and parametrise it?

Unity does not exist, so we choose a cultural standard and scale according to the cultural algorithm. Personally i choose a standard in my own body system and scale  by actively manipulating, touching and measuring against.

Extension and mensuration then are important dynamic defining activities for my sense of "magnitude", amount, bigness,greatness....

So a complex number is actually a polynomial vector, a measuring action ina combined direction in a "plane". Adjugate means  combined by "yoking" together, this is making two separated things work together . The two things are measuring by extending alon ga rule, and rotating around some point to get the measuring in the right direction.

Bombelli was clearly thinking of yoking the √-1 in service with the ordinary numbers, but what he actually did was measure them with a set square during neusis. Bingo! the birth of the model vector!

WEll as i have just explained i think it is the activity of measuring that defines a vector, or rather measuring is a combined vectorial experience in which  statement the use of vector means every which way kind of motion!

The every whichway kind of motions "draw out" lines . areas and volumes in space, in short they boundarise spatial regions.

This i think is precisely what Quatternions do. Firstly they make Volume into a vector by shaping it as a triangular pyramid with the apex at the centre of the unit sphere and the face giving direction or orientation on the surface of the unit sphere. So now we have along with "pointing", "facing" as an orientation notion. Deduced ferom that we have the notion of a surface area vector,which make the the other three sides of the pointing pyramid also surface vectors, and the addition of surface vectors being the "missing"  face of the closed form they make.

On each surface vector we have the pointing vectors that we are familiar with and vector addition is the missing side of the closed shape they make.

On each pointing vector we have scalar addition. Scalar multiplication scales the pointing vectors, squares the area vectors,and cubes the volume vectors.

Within this vector system there is rotational vector development: so interaction of pointing vectors leads to quantized rotation and magnification or stretching; interaction of area vectors leads to quantized rotation and stretching and twisting and magnification, and finally interaction of volume vectors leads to quantized rotation and tumbling,twisting, bending, and skewing and other spatial deformations as well as magnification. In fact i do not know all the outputs for a volume vector interaction these are just what i can presently imagine.

All this and more i think are measurable by quarenion interactions because they hold all 3 levels of vectors. In my view then octonions should do the same but more smoothly and with more detail.

Octonions would kind of represent what i have been thinking of as relativistic motion of a linked system like a tree: each part is linked but independent of the others or rather partially dependent. The utility of these types of measuring and analysis tools i can only imagine, but hey, let the computer sweat the hard stuff!

(http://upload.wikimedia.org/wikipedia/commons/thumb/5/50/RechtwKugeldreieck.svg/356px-RechtwKugeldreieck.svg.png)
Spherical trigonometry (http://en.wikipedia.org/wiki/Spherical_trigonometry) provides a link to quaternions (http://en.wikipedia.org/wiki/Versor).

(http://upload.wikimedia.org/wikipedia/commons/thumb/6/6a/Spherical_triangle.svg/356px-Spherical_triangle.svg.png.)

As you may read, the application of quaternions to special relativity and topology is fundamental. These topics (http://en.wikipedia.org/wiki/Spherical_geometry) have taken on the names of their developers but Hamiltons work with his colleagues and supporters has the seminal pole position!

As i said, despite working on them for the rest of his life Hamilton could not find applications fast enough to dominate the academic market place of ideas, but his contention that quaternions are of fundamental importance has proven to be true.

But i wanted to go down a different track when i started to write so Doug  Sweetser (http://www.quaternions.com/) will have to wait a mo .

 from quaternions i have deduced that we have tensors which amount to pointing vectors, then tensors which amount to area vectorsrequiring 2 pointing vectors and an arc radian  in the form of an isosceles sector to describe them, and finally what amounts to a volume vector requiring 3 area vectors and a spherical triangle area in the form of an isosceles segment (an orange segment) to describe them.

The shapes have been carefully described because they are important as definitions on the unit sphere. For practical purposes and depending on accuraccy required we can use the chord or the tangent.

So for area vectors it is important that they have a rotation motion in their conception, around some point, just as for pointing vectors. This is also fundamental to a volume vector. The point of rotation for a volume vector has a significance in human vision and art, and that is namely perspective.


The stucy of perspective (http://en.wikipedia.org/wiki/Perspective_projection#Perspective_projection) therefore will give insight (http://en.wikipedia.org/wiki/Projective_transformation) into the "workings" (http://en.wikipedia.org/wiki/Projective_geometry) of volume vectors (http://en.wikipedia.org/wiki/Incidence_(geometry)) and provides insight into their interactions (http://en.wikipedia.org/wiki/Projective_plane). 3d animation therefore is an engaging way to study quaternions and their interactions and highlights the range of applicability of Hamiltons discovery.

There are other applications (http://en.wikipedia.org/wiki/Real_projective_plane) i could not think of if i tried!

The Lorentz boost (http://en.wikipedia.org/wiki/Lorentz_boost#Matrix_form) was one thing i had not heard of until recently when Doug wrote the following to me:
ME
> It is satisfying to me that  essentially Grassman's ideas were illuminated
> by Hamilton's quaternions, and because of Hamilton's applications
>  Grassman's obscure writings became slowly appreciated for doing the same
> thing simpler or more easily. and without the non commutativity.
> 
> I can not emphasise how objectionable mathematicians found this property of
> non commutativity. and some work has been done showing that Lewis Carrol was
> so opposed that he incorporated its derision in the Alice stories. The
> success of the Alice stories therefore inveighed against the acceptance of
> non commutativity.
> 
> Hamilton clearly had an insight ahead of his time and if it had been
> embraced Einstein may have not been the first to expound on relativity. The
> development of Tensors etc would have taken a different course as would have
> matrices, all of which are shadows of Quaternions.
 DOUG
There is a fine reason for Einstein and everyone else to ignore
quaternions.  Consider a simple rotation in 3D space.  That is easy to
do with quaternions:
 
R => R' = U R U*
 
where U = (cos (a), I sin (a)), I being a 3-vector.
 
Minkowski argued that special relativity was just a rotation in
spacetime.  The way to write that using quaternions...is missing.  Two
guys in 1910 and 1911 figured out how to do this with biquaternions,
tossing in an extra factor of i, but that is cheating.  Without a
simple way to do boosts, there is NO reason to use quaternions.
Physicists are justified in ignoring quaternions for this one
technical reason alone.
 
In 1995, an Italian fellow named De Leo figured out how to do this
with infinitesimal rotations.  Some college student in Indonesia wrote
me about doing boost, and I recommended the paper.  Thing is, I don't
quite get the paper.  I expected to see hyperbolic sines and cosines,
but they are not there.  In July, I figured it out:
 
R => R' = B R B* + ( (B B R)* - (B* B* R)* )/2
 
where B = (cosh (a), I sinh (a)), I being a 3-vector.
 
Not sure why no one else figured this one out, but it is vital on a
technical level.  All you need is one strike and people will avoid a
tool.
 
ME  
> Scientists, physicists in particular avoided non commutativity by using a
> system that simply ignored those combinations of Hamilton Vectors that
> caused them problems. What you do is show that like Dirac'c Equations the
> odd results have physical meaning. After all Anti matter was ruled out by
> the peers of Dirac at the time.
 DOUG
non-commutativity is essential in angular momentum and quantum
mechanics.  The put it all in their own way, often with the Pauli
matrices.
 
Good luck in your studies,
Doug

There is a taxonomy of plane curves in my mind and an analogous one for curved surfaces.

The plane curve is distinguished by a tool which is a unit circle with a unit tangent. Any curve is measured by this tool and thus classified by whatever modofications are needed to describe it with this tool.

So for example an angle is a curve distinguished from the tangent and lying between it and the circle or intersecting the circle in some chord. Then by degrees i can describe all the polygonal curves until i eventually come to the continuous curves including those that sweep back and intersect the circle like pedal curves etc.. The logarithmic and exponential curves fit nicely in between these dimensions, which if we parametrise will even give a coedinate reference to the curves. Spirals will also fit in nicely

The analogy in 3d a sphere with a tangential plane, allows a classification of the curved surfaces, and of particular interest will be the conic surface and other vorticular surfaces.

I do not know if it can be extended to a classification of volumes, but it would need to include surface distinctions if it can.

Curves that lie entirely within the circle or sphere would form a special group of interest.

What is the fuss about negative numbers and √-1? The greeks based their maths on geometry (http://nrich.maths.org/5961), and the negative numbers make no difference there.

So negative numbers are a different arithmetic object, that had no geoemetrical meaning until Wallis.Even Descartes coordinates did not deal with negative numbers until Wallis.

Bombellis' secret tool was his set square and his neusis, but he tried many methods including 3d models for measurement.

Geometrically -1  is  a square like anyother.

(http://docs.google.com/viewer?pid=bl&srcid=ADGEEShk1BbDeGR3LpHPOvJhWRbm3xT9fkDI9xSIg2JiP-UNUO6jj-sbfSv-xEPSLYPZQLSRxemInG3Uarq03kORKkiXjgADaZ9JgQAf-yOtqd8DVicg8Sf0lTLuTaV2VWMDux2-7D5_&q=cache%3A5FIuFlQiMvcJ%3Ansm1.nsm.iup.edu%2Fgsstoudt%2Fhistory%2Fbombelli%2Fbombelli.pdf%20bombelli%20algebra%20book%204&docid=e21bcf2e423ff5c86e195974a9992f1a&a=bi&pagenumber=13&w=743)

But Bombelli had a way of representing "negative" squares geometrically by placing it under an arm as if in a balance .

 Then usisng his set squares he arranges a system which produces a line or square above the line. This i believe is a positive and negative value system that is geometeical. As you can imagine it required fiddling about a bit until it was just right- this is precisely what neusis mrans!

Now as usual in maths you get to a point where your symbols totally confuse you! What the hell does this mesn you say to yourself and you become lost and confused, but however you plod on and somehow get the right answer!


"In modern notation, Cardano's multiplication was (5−√−15)(5+√−15), and applying the rule for brackets this becomes 25−−15=40."

Bombelli simply observed that the rule of signs still applied to the surd bracket.

Now here comes the red herring! Who knows what a surd bracket means?  Bombeelli did not neither did Cardano or Tagliatelli, and nither for that matter do we.

(http://docs.google.com/viewer?pid=bl&srcid=ADGEESjGXWlpGq6Unmhknk7Kj0ifmNQq49SfDKoIS5l50Krf5u53tXit55yNVn-bSEF-BAvm-_1B9sdGKY5Rei5lJzjqT5jxuf_FUjsYcdeHXemWcgNmgLJ7AmoDFiHnQVB6g_FZHpwN&q=cache%3A0D5bsW3CMmAJ%3Amath.unipa.it%2F~grim%2FPre-mod_algebraQuad11.PDF%20bombelli%20linear%20representation%20powers%20unit&docid=f84b752fbee1a1402f58d837067295a7&a=bi&pagenumber=11&w=743)

What the hell does "radice" mean!? By that i mean, how do you "radice"? Let me simplify the question to get to the point of it--How do you square root?

The "operation" of square rooting, the "algorithm" for finding a root is not taught. We usually skip that and learn them by rote or use tables.

There are geometrical and numerical operations to approximate to a square root. Theodorus spiral is a brilliant example! It exemplifies that geometrically Square rooting is rotating a vector around the circle in a certain way!

Numerical equivalent is to iterate between two values successively "adjusting " the starting point of the calculation and "measuring" how close the "operation" brings you to the desired result.

Whichever way it is looked at the "process"  of radice is a fiddling about operation well described by neusis.

The red herring is : it is not -1 that is the "imaginary" number generator it is the surd operation itself. In short the issue has never really been about number, but about "algorismo" how to proceed in calculating. What Bombelli observed is that it does not matter how you calculate the surd the result still obeys the sign rules if it comes back signed.

Negative numbers were and are the issues in general and square rooting them just highlighted how much western mathematicians with their greek traditions hated them a snot being "arithmoi" that is geometrical objects used as magnitudes for arithmetic.

Bombelli showed that geometrically we can distinguish them by drawing them either side of a line. Wallis later refined this into the number line concept and applied it to an extension of cartesian coordinates to graph the same thing.

How do we square root, cube root nth root in general? we iterate a neusis.

It is the "operation" of neusis that makes the square root of -1 into an operator that rotates the plane! Bombelli did nor have a cartesian plane he had a set square instead, and he rotated it. This has come down to us as rotating the plane, and gone on to develop the notion of relativity through other affine transforms.

Theodorus did the same thing earlier, but o course his ideas were buried by Pythagoras or by his adherents. Archimedes was brave enough to ressurect some of them, and perhaps the most useful of them, and that is why Archimedes is such a master of engineering: he recognised the importance and value of neusis to engineers, despite what the academics were pontificating.

Without neusis it is hard to think how one might deal with many geometrical issues and construction issues. Neusis is after all the greek way of saying "trial and error" through iteration till the error becomes indistinguishable.

This is the fundamental notion of Empiriscism and the methodology of science, and the aim of reductionism;with the meta purpose being to synthesise all the perfect parts into the perfect whole.

 Euclids proposition fourteen (http://aleph0.clarku.edu/~djoyce/java/elements/bookII/propII14.html) I believe is Bombelli's basic neusis as well as his basic "radice" or square rooting algorithm.

I can see Bombelli's set square and i can see the semi circular rotation required in the construction. Thus a rotation by π is needed but the value is read off at π/2 at the specific corner of the rectangle.

This site more than adequately illustrates my meaning of neusis, but more importantly it demonstrates why Geometry was so loved by the Greeks: motion and transformation through motion of specific elements!

When i learned geometry we had a text book, a set square, protractor, and a pair of compass with a marked ruler. We did not have a java applet or even a film to suggest moving the elements of the figures. Those of us who were good at geometry "spotted" the required basic elements of a proof, and naturally moved the book or the diagram around in space or in our imaginations, others who struggled did not move the elements or even the book!
Nobody ever said this is what you were upposed to do, and in fact are rquired to do in some "proofs". Nobody mentioned "neusis"

At primary level or even at secondary level, Nobody mentioned in my day that ruler and compass was in fact frowned on by the "greeks", who desired a "pure reason" for their demonstrations, not an approximation.

Even the Greek academicans were up themselves! Archimedes was an engineer and whatever worked he used! So was Bombelli.

In the development of their abstraction there are many false assumptions that can be pointed out in the greeks "fashionable" derivation of basic theorems, but in this case Neusis is my concern. It was absolutely relied upon until it became so commomplace that it was not referred to, and then it was forgotten and then repudiated as infereior. The fact is that parallel lines are the formalisation of nuesis, along with the circular rotation. Because for everything in Euclids elements one could "replace" neusis with these formal constructions, i think that  is probably why it fell out of favour.

However as the master of spirals observed , not every curve is a circle! Archimedes in fact demonstrated the necessity of retaining neusis in geometry by trisecting the angle easily and elegantly, and then he went on to show how a spiral neusis path could do the same thing.

Sometimes engineers and pragmatists have to rescue mathematics from stuffy academicians, you know. Sometimes you have to give it back to the people to play with! A sentiment that resonates with Bombelli, Duerer and other great communicators and innovators.

It is of inerest to me that in this construction of the square root a type of spiral curve is evident, linking back elegantly to Theodorus. The points of intersection G and E can be used as generators of an Archimedian style spiral.

Note to myself:

Pythagoreans constructed the arithmoi from the proto arithmoi. The geometry of space suggested to them that a unity existed, because they could construct, so it seemed any rectilinear figure  and reduce it by neusis to a rectangle an from there to a square.

Transformations of this sort they believed happened all the time and that was the explanation of the construction of the cosmos.

The special transformations were parallelogram transformations and triangle transformations between parallel lines,plus transformations by rotation in a circle. These were fundamental because the area magnitude was not changed, meaning matter was not created or lost and therefore there must be a fundamental unit that is unchangeable. They called it the "atom" and it was a unity that scaled up to every other form of matter and it was constantly moving through space, and changing thereby through interactions with other portions of matter.

The geometry with its portions and proportions was an exact model of these atoms  it was thought  and the proportioning revealed the ratio of the amounts of atoms, the fundamental unity.

By constructing various rectilinear shapes they found they could transform all to rectangular or square forms and one could explain how unity worked to build portions. They found the proto arithmoi as objects that could not be transformed beyond a simple rectangle. They therefore were proto forms that could be developed into other forms, but were irreducible to other forms.

Like one or unity, the unit square they could be scaled to other forms and so with unity formed the prime elements of the cosmos. Finding them became important to fully describe the geometry of the universe.

It soon became apparent that they scaled, but not independently! They crossed each scaled arrangment in a mesh form

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The structure was used to organise the proportions into arithmoi or scalars, which later mathematicians named as integers and fractions and used them as a unified measuring scale. Geometrical figures or arithmoi had become numbers, and numbers went on to lose their scalar significance, until the development of vectors.

Therefore there exists a fundamental group of geometrical figures for every abstract ring, group, field,equivalence class, or number set.

Descartes demonstrated that the basic operations of number had geometric counterparts, not realising that for the west arithmoi are the root source of the number concept and operations, that is greek geometrical thinking produced the scalars and their operations and Eudoxus organised their application in proportioning arguments , based on the neusis and transformations of geometrical forms.

Gradually over time the pythagorean ideal was eroded to allow for the real and hypereal nymbers, but the fact that they were still scalars was being lost at the same time as vectors were being found as fundamental elements of description.

There is no fundamental basis for unity, only a fundamental scalar role, and the mysterious atom has been fractured enough times to support the notion that infinitesimally small divisions may be possible forever.

Using the Planck constants as unity only re emphasise the scalar nature of measurement systems and the vector basis of notFS,that is the motion field.

Now magnitude is a visual and kinaesthetic sensation, allied to direction and called a vector. However we have other sensations which are allied to direction and should also be understood as vectors" taste, smell and hearing. As these are more directly linked to the brain through specific mesh sensors located regionally in the head and magnitude does not seem appropriate measure, but Intensity does.

Therefore a vector is an entity that has intensity and direction and rotation,or magnitude and direction and rotation.  

So the research comes full circle.

Bombelli's vector model when generalised to 3d produces a vector shape like Hamilton's Quaternions.

So in fashioning a measure i took design cues from a sextant, a theodolite, and a sundial. I could not get the measure to measure within the body of the unit sphere, so i went for a half sphere. Looking at the yin yang symbol i realised i could fix the measure orthogonal/tangential to a sphere shaped round a yin yang symbol. Then i realised that Bombelli had stumbled onto an n dimensional space!

Firstly because he was using a set square he was  utilising the fundamental metric of space and carrying with it trigonometrical relations. Therefore he had everything he needed of the cartesian coordinate system without needing the coordinates. Thus the model vector in its entirety is needed to visualise a vector in 2 space.

In 3 space the model vector is able to rotate freely about a point on one of its corners. The resolution of the model vector is always the hypotenuse.

Now for generalised coordinate vectors the set square is not square, but a general triangle with resolution dependent on the sides of the triangle and the angles between them.

Concentrating on the right triangle it suddenly appears that for each resolution i can choose an orthogonal vector to form with it a resolution2 . Then repeat the process with resolution2 to produce resolution3 and so on. This set of orthogonal vectors are a tangential set of vectors to the curve traced by the resolution vectors. Thus the resolution vectors , the curve and the tangential vectors form an n-dimensional space, in which the tangent n is orthogonal to the space of resolutions n-1 .

Now this system of tangents and resolutions is similar to Theodorus's spiral, and therfore a n dimensional vector space has a tangential envelope which is a spiral form.

By choosing the curves appropriately i could combine n dimension vector spaces to delineate a surface, but i will need to understand better about bundling, nearness of curves, any cross relationships and system constraints or conditions such as Lagrangian or Laplacians. I would need to study Hilbert spaces (http://en.wikipedia.org/wiki/Hilbert_space) to see if this has already been done, but nevertheless i have a correspondence between vortices and spirals and n dimensional orthogonal vector spaces.

My spider friends have been building n dimensional vector spaces for a while it seems and the cone spider shows that they can be used to cover surfaces and volumes.

Why mathematics should be a sub branch of computier science.

I think so because maths is about space and motion in space and relative equilibrium on space,and we no longer have to specify that the objects we play with stay still or follow simple curved paths. In fact we can understand dynamic systems better by watching animations of first analyses. What is always forgotten is that the mathematical geniuses of the past and even today were often prodigious calculators,and could quickly compute variations to formulae or propositions to get numerical confirmation of suspicions. Euler for example loved infinite series!

For those of us not so gifted a calculator or better still a computer puts the same intuitive power in our hands. Therefore Runiter, spacetime, mathmatica, mathmaxima are essential ools worth learning.

If one can learn to programme that is good, but the applications most relevant require a concerted effort from many to be recognised as useful.

Maths online (http://www.math10.com/en/maths-history/history4/Mathematics-Descartes-Fermat.html) giving a historical perspective.We see the growing diffusion of greek scientific methods and ideology, and a move away from arabic rhetoric to symbolic notation.

I find also the theory and definition of "number" (http://www.math10.com/en/maths-history/numbers-counting/numbers.html) arithmoi clearly understood as geometrical objects and being able to measure and scale and be compounds of units.
The essential and mysterious notion of unity is expressed here as existing, existence, being,occupying space and moving.

An account of greek Geometry which makes the mistake of separating mathematics out and so failing to perceive that the whole was the greek science. (http://www.math10.com/en/maths-history/history1/classical-greek-geometry1.html)

We find a link back to an older science of the Sumerian, Egyptians and Dravidians.

"   Around the year 390, Plato visited Sicily, where he came under the influence of Archytas of Tarentum, a follower of the Pythagoreans. Archytas studied, among other mathematical topics, the theory of those means that are associated with Greek mathematics: the arithmetic, geometric and harmonic means. Plato returned to Athens in 388, and in the next twenty years, his Academy came into existence. The purpose of the Academy was to train young people in the sciences (mathematics, music and astronomy) before they undertook careers as legislators and administrators. The two main interests of the Academy were mathematics and dialectic (the Socratic examination of the assumptions made in reasoning). While Plato regarded the study of mathematics as preparatory to the study of dialectic, he nonetheless believed that the study of arithmetic and plane geometry, as well as the geometry of solids, must form the basis of an education leading to knowledge, as opposed to opinion. Plato’s teaching at the Academy was assisted by Theaetetus, whom we have mentioned above. Eudoxus of Cnidus, a pupil of Archytas and an important contributor to the emerging Greek theory of magnitude and number, also taught from time to time at the Academy. Plato’s role in the teaching at the Academy was probably that of an organizer and systematizer, and he may have left the specialist teaching to others. The Academy may be seen as a place where selected sciences were taught and their foundations examined as a mental discipline, the goal being practical wisdom and legislative skill. Clearly, this has relevance to the nature of university learning nowadays, especially as it relates to the conflict between a liberal education, as espoused by Plato, and vocational education with some special aim or skill in mind.

Plato’s enthusiasm for mathematics is described by Eudemus, writing some time after the death of Plato:

• Plato . . .caused the other branches of knowledge to make a very great advance through his earnest zeal about them, and especially geometry: it is very remarkable how he crams his essays throughout with mathematical terms and illustrations, and everywhere tries to arouse an admiration for them in those who embrace the study

of philosophy.

Aristotle (384-322 BCE), the famous philosopher and logician, came to Athens in 367 and became a member of Plato’s Academy. He remained there for twenty years, until Plato’s death in 347. As we noted above, in Plato’s time, dialectic was of primary importance at the Academy, with mathematics an important prerequisite. Aristotle held that the mathematical method then being developed was to be a model for any properly organized science. Greek mathematics at the time was distinguished by its axiomatic method, and sequence of reasoning, from which irrefutable theorems are derived. Aristotle required that any science should proceed as mathematics does, and the mathematical method should be applied to all sciences.

Aristotle is important for laying down the working method for each demonstrative science. Writing in his Posterior Analytics, he says:

• By first principles in each genus I mean those the truth of which it is not possible to prove. What is denoted by the first terms and those derived from them is assumed; but, as regards their existence, this must be assumed for the principles but proved for the rest. Thus what a unit is, what the straight line is, or what a triangle is must be assumed, but the rest must be proved. Now of the premises used in demonstrative sciences some are peculiar to each science and others are common to all . . .Now the things peculiar to the science, the existence of which must be assumed, are the things with reference to which the science investigates the essential attributes, e.g. arithmetic with reference to units, and geometry with reference to points and lines. With these things it is assumed that they exist and that they are of such and such a nature. But with regard to their essential properties, what is assumed is only the meaning of each term employed: thus arithmetic assumes the answer to the question what is meant by ‘odd’ or ‘even’, ‘a square’ or ‘a cube’, and geometry to the question what is meant by ‘the irrational’ or ‘deflection’ or the so-called ‘verging’ to a point.

Aristotle notes that every demonstrative science must proceed from indemonstrable principles; otherwise, the steps of demonstration would be endless. This is especially apparent in mathematics. He discusses the nature of what is an axiom, a definition, a postulate and a hypothesis. It is quite difficult to distinguish between a postulate and a hypothesis. All these terms play a leading role in Euclid’s Elements.

Aristotle’s influence on later European thought was immense. For many centuries,     "

Although the word mathematics is used here liberally, it means science not mathematics. This science as Plato enthused was an analytical, dialectical, deductive mathesis, that is, doctrine of praxis. Of course it was applied to space in motion, and motion in space. It was truely a spaciometry, not just a geometry.

The most detailed account of foundational indian science (http://www.math10.com/en/maths-history/math-history-in-india/mathematical_achievements_of_aryabhatta.html) i have read so far.

The information highlights the rhetoric of early thinkers, which put another way is using words to stand for things, relations, operations and aspects and not the customary referent. The other way of thinking of rhetoric is analogous thinking, or metaphor. In this sense there is a connection with Chinese I ching and Nine Chapter formulations .

It is clear that Indian science is different to greek science, preferring to use all aspects and attributes of form in their rhetoric. and enjoying the relationships and poetry of forms in their exposition or exegesis. Thus the rhythm and metre, the arrangement and juxtaposition of rhetoric conveys a major part of the sense of the "advice". Like a song the advice relates analogous things regardless of context, and so meaning of the "advice" can be found in all sorts of contexts. This is the true heart of Algebra, a generality of applicability, an essence of relation.

The rhythmical and metrical nature of indian science refers directly to the process of iteration. Of all the fundamental things we know of being human, and existing in a world outside of human subjectivity, iteration has got to be the one fundamental common action.

While mathematics has lately distinguished itself by complex notation, this has been a move by people who tired of the constant repetition. But the indians enjoyed the rhythm of the repetition and so carried the relationships in hymns and and songs and poems. (http://docs.google.com/viewer?a=v&q=cache:zzIQGcb9YiUJ:www.millersville.edu/~cardwell/fall09/310-01/Ancient%2520Indian%2520Math%2520-%2520GSS%2520-%25202March05.pdf+indian+geometry&hl=en&pid=bl&srcid=ADGEESjZksEiauyXr9c6rjPLVGoQeGognAlVDW5BWCfyzNHH_bFU418-AthVbKKsjTzuz99MvIPbNkrlQy_5IBG6PTJhzYgn5HayGAOlNuRtzTxBKYvLhL48NbW5wQg4WAQUVf-kUSTE&sig=AHIEtbTngrB-2lco_4wQxKz1rs_tF1dtXQ). This is markedly different to the greek discoursive dialectical style, and is in many ways the source of the clash between greek and indian science.  

Technical treatment (http://www.stanford.edu/class/history34q/readings/Rotman/Rotman_Metasubject.html) of number. Dealing with this metaphysical analysis (http://books.google.com/books?id=Z2MVAAAAIAAJ&pg=PA444&lpg=PA444&dq=greek+geometry+arithmoi&source=bl&ots=1Ar5LWr3cJ&sig=7MAU4leHTaaEwUQxt5khnoK5D1s&hl=en&ei=vtP-TPreHdiAhAeFqt3xCw&sa=X&oi=book_result&ct=result&resnum=4&ved=0CBwQ6AEwAw#v=onepage&q=greek%20geometry%20arithmoi&f=false) .  Again it is a science not a mathematics that the greeks were exploring, a science of space and motion. Arithmoi

In short arithmoi are scalars of a unity. You may choose the unity, and all its properties are scaled. However each unity consists of space which is part of unity and not a scalar. This part can be proportioned but not represented by arithmoi scalars .

Thus arithmoi are an attribute to space which i as  a animate attribute and having attributed proceed to count, scale and manipulate. However space itself within these confines is also being manipulated. Arithmoi enable me to scale space,so when the arithmoi become a mouthful i can rescale and calculate at that new scale. Thus the greeks had self similaity built into their scalars.

Their analysis of the space within a unit attributed to it divisibility and aggregation,but no mensuration. However by changing scale for unity mensuration could be effected, but the same condition applied to the space within the unity. Therefore not only did they appreciate self similarity they had the apriori of iteration. Thus the greek idea of space was necessarily nested abd fractal.

The Platonic notion of space was that it was reducible to two fundamentals: unity and extensible/elastic magnitude.  Pythagoras had thought that there was a fundamental unity which by definition was fixed and merely scalable upwards and thus was the measure of all things.

A nice treatment of indian sensibilities in their science (http://hubpages.com/hub/ANCIENT-INDIAN-MATHEMATICS)

Brahmagupta the man who invented negative numbers.

"I......n the Brahmasphutasiddhanta he defined zero as the result of subtracting a number from itself. He gave some properties as follows:-

When zero is added to a number or subtracted from a number, the number remains unchanged; and a number multiplied by zero becomes zero.

He also gives arithmetical rules in terms of fortunes (positive numbers) and debts (negative numbers):-

A debt minus zero is a debt.

A fortune minus zero is a fortune.

Zero minus zero is a zero.

A debt subtracted from zero is a fortune.

A fortune subtracted from zero is a debt.

The product of zero multiplied by a debt or fortune is zero.

The product of zero multipliedby zero is zero.

The product or quotient of two fortunes is one fortune.

The product or quotient of two debts is one fortune.

The product or quotient of a debt and a fortune is a debt.

The product or quotient of a fortune and a debt is a debt......."

Brahmagupta (http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Brahmagupta.html) wroote a hymn to his goddess in which the void was no longer excluded, but reverenced as the opening or unfolding of the universe. His attribution of debt or fortune to numbers is a worshipful insight into the hidden world of his goddess. This kind of attribution was not strange or uncommon in indian sensibilities, as they understood unity to be in and of itself, the qualities and attributes that the space utilised to standardise a measure has. Everything attributed to a unity was a referent to that unity and is scalable with that unity.

So in his consideration of the Astronomy of his time and the origin of the  universe he considered the unfolding from the void as the ultimate source of everything,but true to indian philosophy of astrology it unfolded good fortune and bad fortune first in equal measure. This is the meaning of fortune and debt, and why negative numbers are so hated!

Out of respect for Brahmagupta and his goddess it is probably best to explain what he was about.

BG defined unity in the usual customary way in india, replete with all its attributes and potentials. Any advice he would give would have to have applicability in many analogous situations and so it was usual not to constrain a definition unecessarily. The greeks however did not obseve such niceties,attempting instead to grasp the robust core of the essential nature of space. they left a lot of things out.

Having defined unity, in which no measurement could be made because it itself was the thing, the space by which measurement was made, BG observed that unity could be taken from itself to leave the void. No one had done this before because it is understood that unity and all measurement are relative to each other in scalar ratio forms, and all things spatial exist in a form.

That form no matter how small is a unity, thus to have no unity implies that it does not exist. The idea that all is nothingness is neither a greek or indian or ancient idea. In fact although i once thought it was a medieval christian idea it is not a serious idea in any culture, except unphilosophical scientists! They of course blame Newton,but that does not hold up as Newton was a Descartesian.

So for BG unity "offing" itself was not "annihilation" introducing nothingness, but returning to the infinite void from which all things came, and in which is all potentiality in indistinguishable form and activity. The void and therefore 0 was a dynamic state of superpositional potential, anything could happen , and anything could explode out of the void, and frequently did!

So having made what seemed an innocuous observation that unity "subtract" itself returns to the void that is 0 BG advises how to use this entity with regard to our custumary manipulations of unity. The customary manipulations of Arithmoi became called arithmetic, and arithmoi strictly were the integers/scalars  above unity. If our unity was a calculus(stone) then our customary manipulations would be called calculation. in any case ar(ea) were the unity of the greeks and the ar(ithmoi) were geometrical shapes and volumes based on those and arithmetic is the manipulation of the ars(areas,volumes)

So what was this new observation to be called? BG may have given it a name, but it like unity was a separate idea to the scalars.

Today we have a candidate name called nullity and infinity, and some axioms based on it. BG saw nullity as a relationship with unity which defined the meaning of our measure and our operators. When doing everyday calculations our operators have one meaning, but when dealing with nullity and unity our operators take on a subtly different meaning.

To start with how can one take unity from itself? you can make a relative motion of unity to another place but unity still exists, just in another place: by this BG advises on relativity.

The only way was if unity was returned to the void by an equal but opposite unity which exists for that purpose: by this BG advises on quantum phenomena.

BG as an example decomposes the void by an astrological measure: fortune and poverty. AS i say this is an example of its applicability that BG advances. He expects fully that others would draw out the other applications, and there are probably others he had in mind, but by aligning his thinking with the astrologers he fulfilled his other hat as a keeper of the astrological knowledge of indian astronomy.

".....Brahmagupta's understanding of the number systems went far beyond that of others of the period. In the Brahmasphutasiddhanta he defined zero as the result of subtracting a number from itself. He gave some properties as follows:-

When zero is added to a number or subtracted from a number, the number remains unchanged; and a number multiplied by zero becomes zero.

He also gives arithmetical rules in terms of fortunes (positive numbers) and debts (negative numbers):-

A debt minus zero is a debt.
A fortune minus zero is a fortune.
Zero minus zero is a zero.
A debt subtracted from zero is a fortune.
A fortune subtracted from zero is a debt.
The product of zero multiplied by a debt or fortune is zero.
The product of zero multipliedby zero is zero.
The product or quotient of two fortunes is one fortune.
The product or quotient of two debts is one fortune.
The product or quotient of a debt and a fortune is a debt.
The product or quotient of a fortune and a debt is a debt.

Brahmagupta then tried to extend arithmetic to include division by zero:-

Positive or negative numbers when divided by zero is a fraction the zero as denominator.
Zero divided by negative or positive numbers is either zero or is expressed as a fraction with zero as numerator and the finite quantity as denominator.
Zero divided by zero is zero.

Really Brahmagupta is saying very little when he suggests that n divided by zero is n/0. He is certainly wrong when he then claims that zero divided by zero is zero. However it is a brilliant attempt to extend arithmetic to negative numbers and zero...."

Actually BG was right, because he did not make the mistake of thinking that unity and nullity were arithmoi to coin a greek phrase.

What they are we have only just begun to realise: they are seeds of all our measurement and manipulation and interpretation of a dynamic space, a motion field if you will.

Some fundamental relations:

1-1=0  additive inverse /diffusion
1/1=1  multiplicative inverse/ condensation
cos^2+sin^2=1^2  rotational equilibrium
√-1*√-1=-1  dynamic equilibrium

e^(π*√-1)+1=0 rotational dynamic equilibrium(?) observer
0/0=0
n÷0=n/0  - ratios or fractions of the void /uniqueness of relativity
circumference: radius =2π : 1 radian measure.

ø*√-1=ln(cosø+√-1*sinø) rotational dynamic equilibrium(?)subjective

Having no fear of astrological misfortune, and neither impeaching or besmirching any god, i gaze steadily at the void and the decompositon  of it that BG has brought to my attention.

Firstly the attribution of plus and minus to a magnitude, referring to the fact that independent of our involvement, should such attributed magnitudes come into contact they would automatically subtract and leave a balance.By this BG advises us of a ceaseless activity in space that is computational.

However at present there seems to be no automatic emission or decomposition of the void into these components in what BG has formulated, at least by report.

However i do not worry, as this is a new apprehending of his insight and over "time" more will flower from it.

Now we may also look for other attributes like debt and fortune,+ and - which are of this bipolar structure and which give a "zero sum"  in some sense. Dynamic equilibria as well as static equilibria are fruitful candidates.

Concentrating briefly on +1 and -1 as unities, the BG rules imply that all transformations in one apply equally to the other scalar arithmoi, the spaciometry of forms and their manipulation and transformation- in a simplified set of operation equivalent to an arithmetic.

Where we do have to nake alteration and be mindful is whre the two unities mix, and the rule arelaid out clearly there: -+=-;  +-=-.

So bearing this in mind we have no trouble determining√-1.

Firstly √ has been defined as the ± of the root of the magnitude. A tautology is used in this definition which i think is simpler expressed as: the magnitude is without sign and is a scalar of unity. To this scalar we attribute sign as necessary. The rules then are rules of how we attribute sign before and after calculation.

That being said we have been defective in our definition of √. We have defined ± as plus  or minus but have not defines plus AND minus +/-. As you can see i have had to switch them about when ±would more naturally mean plus and minus.

The difference this makes is simple : √-1 can now be defined consistently as plus and minus 1 (±1)

Where √+1 is now defined as +/-1(plus or minus 1).

These 2 definitions represent a dynamic equilibrium and 2 static equilibria.

Shunaya (http://en.wikipedia.org/wiki/Śūnyatā) simply put means replete/swollen with every conceivable and non conceivable attribute.

So when Bhudda says the world is empty of self, he advises not to pursue self ,or that which does not exist independently. As all things are dependent, he advises to seek all things, a wider apprehension. By these advices he directs any listener to a life of active learning and appreciation of all things as an interdependent connected web.

Brahmagupta therefore invites a meditation on the source of all unities by introducing shunaya into our basic conceptions of unity and their scalar arithmetic, or manipulation,or calculation.

There is one thing Brahmagupta advises that is overlooked, and it ought not to be overlooked, and that is the yoke he so deftly and lightly lays on the shoulders of these attributable unities. As light as it is yet it binds stronger than death quantiies and magnitudes in proportions of exactness, in relations of cunjugacy, adjugate companionships and bilateral even multilateral activity and actions and behaviours. Thus what we see intimated is merely the tip of a far more extensive range of attributable properties and decomposition of a very swollen void.

Without shunaya we would not have the Cartesian Algebra as it is extended to today, neither affine transforms or tensor analysis,nor vectors in any recognisable form.

Plato looked at science (Mathematikos) which was concerned with spaciometry, the measurement of dynamic space, and which hoped to apprehend the construction of the universe by the gods, if they existed.and whether they cared or not if they existed. Men (http://www.ellopos.net/elpenor/koinonia/topic.asp?TOPIC_ID=298)(thinkers and manipulators and measurers) existed      anthropoi (http://engforum.pravda.ru/showthread.php?218529-%CE%91%CE%9D%CE%98%CE%A1%CE%A9%CE%A0%CE%9F%CE%A3-Anthropos-Human) they believed to the Theoi (http://www.myetymology.com/proto-indo-european/dyeu-.html) (dharma (http://en.wikipedia.org/wiki/Dharma_(Buddhism)), Dyaus (http://en.wikipedia.org/wiki/Deva_(Hinduism)) sanskrit) that lived above mount Olympus, in the visible theos which is the ouranos, the heavens, and greek men dared to challenge the Theoi the children of Dyaus/Zeus if they could be bothered to fight! One day greek men would sit among the Theoi, or pull them down trying!

Plato looked upon all this in mathematikos (science) and contrasted it with all other forms of knowledge (epistemologia) and declared to Lovers of Sophia, Philosophers that the greatest wisdom, the gift of Sophia was in Science, and that all epistomologia (knowledges) also gifts of Sophia should get off its backside and organise itself like science. He defended that opinion for the rest of his life .

Symmetry (http://epistemologia.zoomblog.com/archivo/2007/11/28/symmetry-breaking-in-a-philosophical-c.html) is involved in the yoke of BrahamaGupta but along with symmetry comes that which is not symmetry, and these stand side by side as a decomposition of spaciometry. It is this decomposition that describes or is described by the fundamental group structure of the void.

As a representation of this i posit the relationship between the sphere and the spherical vortices that constitute the vortex torus form, or as found in oceanic forms the shell vortices. These shell vortices have always been valued as precious objects and beads, but of course there is a time of human exploitation using these precious items,to further imperial aim!

The relationships between spherical space and spherical vorticular space is the relationship between equilibrium, and dynamic equilibrium and runaway disequilibrium leading to boom or bust, bang or crunch!

Brahmagupta advises on symmetrical yokes, but also on non symmetrical ones: n/0 and 0/n.

What 0/0 advises beyond all voids diffuse/evaporate into 1 void( or rather a supervoid which generates voids- in other words a nested fractal)  i have yet to discover.

If I was to say that shunaya meant dynamic space, what would you think? Nothingness? Hardly! You may think "wind", and indeed the Arab sifr does connote that. From this through various ethnological shenanigans we end up with zero. Confusion or what?

Now I mentioned dharma in conection with Zeus and I know this is a misconception, but it resonates so aim leaving it. I could not correctly recall the Sanskrit devah at the time.

Dharma is after all "the rules of perceiving" and that is very relevant to any science.

I have just seen a new banner logo : infinity is visible. Of course n/0 says that in a more cryptic way. So I guess 0/n means that infinity can be portioned and proportionate, where n/0 means visibility is due to disproportionate proportions,ie symmetry breaking. Finally 0/0 must mean "the possibilities are... Infinite"?

We sense, therefore we exist.
We perceive therefore our existence is distinguishable.
We distinguish sensors, therefore there is something to sense.
Because we sense and perceive and distinguish through and with sensors then an external internal sensory decomposition exists.
 The decomposition of sensory data into internal and external components will be distinguished as space, therefore space exists, and has attributable properties, the most fundamental being internal and external to us.

"  Origin and development of the concept of emptiness( wikipedia)

The theme of emptiness (śūnyatā) emerged from the Buddhist doctrines of the nonexistence of the self (Pāli: anatta, Sanskrit: anātman) and dependent arising (Pāli: paticcasamuppada, Sanskrit: pratītyasamutpāda). The Suñña Sutta,[6] part of the Pāli canon, relates that the monk Ānanda, Buddha's attendant asked,

 "It is said that the world is empty, the world is empty, lord. In what respect is it said that the world is empty?" The Buddha replied, "Insofar as it is empty of a self or of anything pertaining to a self: Thus it is said, Ānanda, that the world is empty."

He goes on to explain that what is meant by "the world" is the six sense media and their objects, and elsewhere says that to theorize about something beyond this realm of experience would put one to grief."

Now suppose that the word empty is translated "swollen" or "replete".

Bhudda therefore advises one to go beyond the senses to comprehend the world, In other words be wholly empirical, and therefore dynamical, livng and experiencing each fleeting moment, from moment to moment, And any conclusions you draw from that experience must fail and turn to dust with you,for they are as dynamic and fleeting as you are. Shunaya therefore has the curious attribute of being self reflexive, making sense in all its interpretations.

Brahama gupta is said to have drawn a perpetual motion machine, a wheel based design, therefore a yoke of symmetry, BG thereby advises not perpetual motion of the wheel but perpetual moion of the void, a dynamic void, for it is not a wheel but a symbol of the void he drew. `in this way BG extends his operator beyond scalar representation in arithmetic to geometrical representation in spaciometry. No doubt he applies it to the sphere.

The perpetual rotation of the yoked decompositions we are more familiar with in the sin and cosine version and the Euler Cotes formula

 where represent the constant rotational dynamism.

This of course would not effect the perfect circle or sphere as the addition of its elements are zero!

As in the decomposition into + and minus there is a decomposition into clockwise and anticlockwise, and these are the effective constituent elements of a rotational machine. These i think would necessarily be spiral, and components would be trochoids of spirals. Both cannot exist in he same machine if it is to work, and therefore work has to be done to establish this state of affairs. Therefore we cannot get more out of a machine than we put in, unless we break the symmetry yoke in the system. In oing so the system will explode.

Newton observed, but few have fully understood that all forces we investigate  are equilibrium forces, and arise only in respect to matters of equilibrium. However there are disequilibrium states, but whether they are nested in a larger equilibrium system etc is only a guess at this momemt.

The decomposition of the void is the natural subject of the philosophy of the I ching. For this subject chinese philosophers have devised maps or flow chart like diagrams called Taijitu.

Lai Zhida it is said introduced the taijitu into the yiology of the I ching in 15th century. In doing so he was introducing a "modern" scientific revision of the Yi Ching. 5 centuries earlier Zhou Dunyi (http://en.wikipedia.org/wiki/Zhou_Dun-yi) introduced a version called  the Taijitu Shuo which applied it more generally,and particularly to human activity in all its forms.


Zhu XI (http://en.wikipedia.org/wiki/Zhu_Xi) particularly established this philosophical discipline.

According to the chinese Fu Xi (http://en.wikipedia.org/wiki/Fu_Xi) is the originator of the Yi Ching, and we see here the spoked circle, the twisted vortex and antivortex and the cone out of which Nuwa and Fu Xi emerge, holding symbols of the heavenly powers.(http://upload.wikimedia.org/wikipedia/en/d/d0/Nuwa%26fuxi.jpg)

The development of the Yi Ching (http://en.wikipedia.org/wiki/I_Ching) continues through philosophers and astrologers to apply a divination method to more and more contexts.

Although dense and rhetorically complex the Yi Ching is basically simple. 64 Hexagrams have over time been established as the reference frame. This reference frame is a human measuring instrument of possiblities from the void, and as such represent a decomposition of the void into these 64 distinct groupings.

Thus the void is full of all potential but only 64 are studied by the Yi Ching.

The fundamental decomposition of the void in China  has always been accepted as yoked opposites, in this case Yin and Yang. It is my consideration that this decomposition influence BrahmaGupta, and in fact the Chinese Yi Ching was a curiosity of his in the 6th century AD around the time of chinese indian cultural exchange (http://www.southasiaanalysis.org/%5Cpapers33%5Cpaper3245.html).

Using this fundamental yoked pair as a guide to each line of the hexagram, some activity is used to "draw" decompositions from the void, (http://en.wikipedia.org/wiki/Divination_with_I_Ching) to Realise a supepositional state! However, it was thought that one realisation was not sufficient only giving one aspect of the superpositional state, so over time 6 lines were developed, and thus called the hexagram. As stated 64 hexagrams were developed as sufficient and necessary.

For each of the 6 lines in a hexagram the yin yang decomposition would have either a static or dynamic resolution! so the yoked pairs were recognised as being equilibria that are static and/or dynamic, and decomposed into anticlockwise and clockwise dynamic equilibria and original and reflected static equilibria: thus 4 states.

In terms of the Yi Ching this gives 64*6*4 =1536 different state readings and 16*6*64 exegetical interpretations for a pair of readings and 4^3*6 *64 for any triple of readings etc..

This was felt sufficiently complex for human affairs!

Zhou xi and Zhou Dunyi developed aspects of this traditon to give a sufficiently robust description of probable outcomes to Government decisons, so in that sense it was used as a kind of game theory.

The Yi ching has clear application to quantum chromo dynamics and other quantum descriptions.

In regard to Brahmagupta, i see that considering these things lead to his advice on decomposing the void, and brought from chinese influence the notion of shunaya into indian philosophy in 6th century, through a long cultural contact starting in 4th century?.

Of magic squares, magic circles, the Yi Ching and the Yin Yang as decompositions of the void into yoked unities, the concept of which has given birth to Taijitu of the fundamental process of the universe.

Of how these elements have been algorithmically combined by the Yi ching and BrahmaGupta into the basis of Science from Quantun Chromodynamics to the vast reaches of the known universe expounded in Relativity theory.

TBC.

Of cradles of civilisation (http://en.wikipedia.org/wiki/Cradle_of_civilization) and the mutual development of asian science (http://docs.google.com/viewer?a=v&q=cache:MtukovJfRswJ:stallion.abac.peachnet.edu/humhtm/senapati/HUMN2221/PDF6/Chap5.pdf+chinese+and+indian+civilization&hl=en&pid=bl&srcid=ADGEEShw7wy6cEN1w95PNCk8eHR-SFukNFQLu5jGipVfK9jLfCr10TVnEEQsrTE_NZW0-cVNjiuM-yCAvognxc2vEVfzNoiV4spy98HZiLn5oEFejWJty2hLISUm87fhF2pSQmbeiokR&sig=AHIEtbT9vMS_ilYuKLLEGl9JogMbX3abnA) and near eastern science and the western offshoot through greece. A cultural twinship (http://kolkata.chineseconsulate.org/eng/zlgxw/t676806.htm) against the aryan invasian theory (http://docs.google.com/viewer?a=v&q=cache:-3NiGVzMKvQJ:occawlonline.pearsoned.com/bookbind/pubbooks/stearns_awl/medialib/IM/ch03.pdf+chinese+and+indian+civilization&hl=en&pid=bl&srcid=ADGEESiGuFKkIbSa40kU_XMFUpa5ugxuIihSAk6rXVjzwTZZW3pxx0T-kRjdWMLp69Vpsde0DUcR6H19ewhBjoPRVkD6cF2knDPihxzWgrerPlBgwUG6kUxKgq-QITZxqcdgO3aA05Gq&sig=AHIEtbT957l1LeuRlqGrp88QyGHzFt3Efw) and for the out of india theory (http://docs.google.com/viewer?a=v&q=cache:f5bgeBB70r8J:people.cohums.ohio-state.edu/hathaway24/181indiachina.ppt+chinese+and+indian+civilization&hl=en&pid=bl&srcid=ADGEESj-RaoZtDrbi97pi7VluZvHKLdI09d5SCxlzRFJNZzZvfgi3NPk9SDKM_5D6VXvcGOPYBxeBgmI5nBeVTGzhLJedqT5uPB3vTnDVXWXBKXQNW972Geq8-Jq-VTIQ7AxnvL8ed7T&sig=AHIEtbRqv5-Tre9Lyh73PltLSWR2b-eebg)

Civilisation (http://au.answers.yahoo.com/question/index?qid=20100310185736AApUauG) and human presence not the same.

The pragmatism of Confucianism (http://en.wikipedia.org/wiki/Confucianism) created a stable void in personal thought which the Buddha and Buddhism filled appropriately. The taijitu expressed this fusion of inner striving and outer order and harmony. Through the bridge of Buddhism the chinese had access to Hinduism (http://en.wikipedia.org/wiki/Hindu_philosophy) and it s influence on Indian thought and practice. in the fusion with buddhism.

"Samkhya
Main article: Samkhya

Samkhya is the oldest of the orthodox philosophical systems in Hinduism. Samkhya is a strongly dualistic philosophy that postulates everything in reality stems from purusha (Sanskrit: पुरुष, self, atma or soul) and prakriti (matter, creative agency or energy). There are many living souls (Jeevatmas) and they possess consciousness. Prakriti consists of three dispositions known as qualities (gunas): activity (rajas), inactivity (tamas) and steadiness (sattva) which arises when the two other gunas are held in equilibrium. Because of the intertwined relationship between the soul and these dispositions, an imbalance in disposition causes the world to evolve. Liberation of the soul happens when it realizes that it is above and beyond these three dispositions. Samkhya denies the existence of God.[2] Western dualism deals with the distinction between the mind and the body,[3] whereas in Samkhya it is between the soul and matter.[4] The concept of the atma (soul) is different from the concept of the mind. Soul is absolute reality that is all-pervasive, eternal, indivisible, attributeless, pure consciousness. It is non-matter and is beyond intellect. Originally, Samkhya was not theistic, but in confluence with Yoga it developed a theistic variant.  "

The gunas correspond to motions and distinguish 3 sorts of motion: acceleration(rajas), static equilibrium(tamas), dynamic equilibrium(sattva). Though dualist the chinese notion of yin and yang (yoked pairs) is not present in indian's oldest philosophy as cited here, therefore i venture that yoked pairs is a unique to chinese philosophical idea that influenced later Indian thinking in the person of Brahmagupta.

Of chinese science/astrology and quantum chromo dynamics.


Hinduism is like saying Britishism or Americanism: it refers to a geopolitical regional entity not a faith. Buddhism again refers to a belief system based on "buddha's" rules of perception, and as such is a container of any cultural ideas of those who apply the rules of perception, the Dharma. Indian philosophical thoughts in all there varieties were thus brought into contact with chinese philosophical pragmatism or Confucianism.

Confucianism again is a carefully constructed container like a "dharma" for behaviour not perception, thus Buddhism and confucianism actually complement and attract each other.

Through this connective tube of buddhism and confucianism indian philosophies flowed into china and chinese pragmatic philosophies flowed into india. Thus Brahmagupta came to examine chinese science and astronomy and the Yi Ching. AT about the same time Lai Zhide examined indian science astronomy and astrology. Lai Zhide revised the Yi ching to put it on a more astronomical, astrological basis, that is a more scientific basis and redesigned the Taijitu accordingly. Zhou Dunyi revered the traditional taijitu in his later combination of the buddhist, confucianist taoist  streams into neo confucianism.

Brama gupta in studying idea from the Yi Ching was lead to ad to the 3 motions of Shunaya the notion of the yin and the yang, a balance that was yoked and could perform the 3 motions in a yoked form.

Fro this Brahmagupta derived the rules of yoked motion. These rules are and always have been a meditation and a sudhanta,

From india through the arab empire it spread throughout the world and impacted on everything. Through it the world has derived the so called "complex numbers", but which Bombelli called "adjugate" to numbers, that is yoked, and he recognised the pairs that he called "conjugate": the yoked pairs.

Through the Bombelli operator the rest is history because we end up at Quantum Chromo Dynamics through a wonderful and curious route.

However as i have pointed out there never really was a problem with the square root of -1, rather it was a problem with misconceptions of Brahmagupta's sudhanta.

For me the √-1 is simply +AND- 1.

Shunaya means the void which i instinctively know is magnitude. However magnitude is also instinctively a measure but what measure?

I have to explore the concept of measuring, the activity i engage in voluntarily and involuntarily and identifiably as measuring. To do this i have to start with an axiomatic model of measuring.

Indian concept of oneness (http://www.google.com/search?q=indian+concept+of+oneness+eka&btnG=Search&hl=en&client=opera&rls=en&sa=2).

The Indian concept of shunaya has a deep meditative role in mathematics for indians , but the concept of 1 Eka  has a transitional role, that is it transitions from the individual to the whole as a universe,

Thus the concept of unity (http://docs.google.com/viewer?a=v&q=cache:5N4o4XYU5HUJ:ocw.mit.edu/courses/special-programs/sp-2h3-ancient-philosophy-and-mathematics-fall-2009/assignments/MITSP_2H3F09_ses5.pdf+greek+measure+unity+one&hl=en&pid=bl&srcid=ADGEESiZU56aTIBMx6_M2E138znOXqromIMKKAMdYiNIzQsv1GDXfJKP856BBCfrhZKKwasDTFIypoc5GBTLq2B5UNqaobkBP9DMzXTzjUdTxZnugzXz3xi7hQfEWUCtvbh0JgN5YxDa&sig=AHIEtbSGyx_lP9o_WqSWYBIa0ka9lM0Eqg) as a meditative  search into the meaning of the universe is entirely greek!  So the indians studied shunaya, the greeks unity (http://muse.jhu.edu/login?uri=/journals/journal_of_the_history_of_philosophy/v013/13.2pancheri.pdf) monos.

The earlier greek philosophers (http://personal.bellevuecollege.edu/wpayne/presocratics.htm) meditated on the world, the universe and came up with "unit" as the basis , The indians came up with shunaya

Man is the measure of all things (http://employees.oneonta.edu/farberas/arth/ARTH200/Body/man_measure.html) is 0/n.

"NUMBER VS. MAGNITUDE (http://www.learner.org/courses/mathilluminated/units/3/textbook/02.php)
In the mathematics of early Greece, there was a strong distinction between discrete and continuous measurement.
Number refers to a discrete collection of atom-like units.
Magnitude refers to something that is continuous and that can be infinitely subdivided.
Rational numbers can be expressed as decimals that repeat to infinity.

Early Greek mathematicians divided mathematics into the study of number, or multitude, and the study of geometry, or magnitude. The multitude concept presented numbers as collections of discrete units, rather like indivisible atoms. Magnitudes, on the other hand, are continuous and infinitely divisible. Because length is a magnitude, a line segment can be divided as many times as one likes. The Pythagoreans believed that magnitudes could always be measured using whole numbers, which would imply that lengths are not infinitely divisible. Other schools, such as the followers of Parmenides, known as the Eleatics, believed in the infinite divisibility of magnitudes.

Parmenides taught that true "being" is unity, static, and unchangeable. This is similar to the idea that "all is one," which implies that concepts such as multiplicity and motion are illusions. If everything is part of the same thing, then there are no "multiple" things and, consequently, no motion, which is the change in position of one thing relative to another. Pythagoreans believed in multitude and motion perhaps because these concepts are intuitive, part of collective common experience. A consequence of the Pythagorean notion of multiplicity is that magnitudes should be commensurable. To the Pythagoreans, the idea that between any two quantities in nature there exists a common unit of measure, a common denominator, may have been comforting. It perhaps suggested that the rational mind can always find a solid basis for comparison, and does not have to rely on guesswork to say definite things about reality.

It would be easy to dismiss the Eleatic view, if it were not for the arguments of one of Parmenides' most famous pupils, Zeno. As we shall see, Zeno argued against the Pythagorean notions of multiplicity and motion, using infinity to show contradictions in this view. Prior to Zeno, however, problems with the Pythagorean viewpoint arose from within their own ranks in the form of an independent thinker by the name of Hipassus of Metapontum. Hipassus showed that magnitudes are not always commensurable, an idea that upset his peers to such a degree that, as the legend goes, he was drowned for his heresy. In the next section, we shall examine the idea and consequences of incommensurable magnitudes.:"

a lexicon (http://books.google.com/books?id=UmbRAAAAMAAJ&pg=PA641&lpg=PA641&dq=unit+/atom+lexicon+greek&source=bl&ots=H40bUb0Fgh&sig=ZXj9fA2O1hHFxh_xtZmv2WjExqE&hl=en&ei=-koJTdqdMIOWhQfepsCrDw&sa=X&oi=book_result&ct=result&resnum=1&ved=0CBIQ6AEwAA#v=onepage&q=unit&f=false)

mona- and mono- (http://books.google.com/books?id=UmbRAAAAMAAJ&pg=PA641&lpg=PA641&dq=unit+/atom+lexicon+greek&source=bl&ots=H40bUb0Fgh&sig=ZXj9fA2O1hHFxh_xtZmv2WjExqE&hl=en&ei=-koJTdqdMIOWhQfepsCrDw&sa=X&oi=book_result&ct=result&resnum=1&ved=0CBIQ6AEwAA#v=onepage&q=unit&f=false) unit/single. page 331 and page 613. Entos is more unity by inclusion (http://www.archive.org/stream/significantetymo00mitcuoft/significantetymo00mitcuoft_djvu.txt).But i have at last found the common counting (http://phrontistery.info/numbers.html) namers in greek and en is "one" even if it is written α (alpha). En of course also means in, within, so the idea of wholeness being inclusive is apparent.

Before i leave this i point out that greek thought was based on units (http://en.wikipedia.org/wiki/Monad_(Greek_philosophy)), that could be established at any scale, so the pythagorean notion of being able to define a unit that would measure any magnitude is intuitively correct. So what Hipassus and Theodorus showed is that irrespective of what unity you choose there will always be magnitudes that cannot be measured. Both i would say suffered the same alleged fate. Eudoxus covered over the crack by positing a theory of proportion, which satisfied all that proportioning and reasoning could always be done and so it did not matter as these magnitudes themselves could be used as unities. Therefore from the time of eudoxus it has always been known that arithmoi are scalars and that all proportions and ratios are essentially holding information about scale, All manipulation is manipulation of scales and rules found true for scalarswould universally apply. Thus there was no need to actually see an atom as the scaled up versions exemplify their behaviour.

Atoms accordingly would have different sizes and these sizes would form fundanental unities like the proto arithmoi. What these atomic magnitudes were was the focus of their continued research, to no avail. until the unity scalars could be arrange conveniently and themselves ino a scale of scalars, the continuous fractions in use t the time was tedious to the greek mind, however it was beloved by the indians like a rhythmical song.

Of Cantor and shunaya.

Cantor in his exploration of sets came across the infinite sets, and on exploring them came face to face with shunaya. It so unsettled him that he thought he had sinned against god and was going mad! after recovering he left us with the notion that there are different kinds of infinity. This is what Brahmagupta meant when he wrote n/0 the countable fractions of infinty!

Cantor through modern number concepts added the notion of uncountable infinity, but NOT different magnitudes of infinity. There is only one "magnitude" to "infinity" or shunaya and that is shunaya.

BG used an integer n when he expressed countable fractions of infinity. If we use a real number r then r/0 represents uncountable fractions of infinity. These scalars are intuitively (mod 0) and form a (mod 0) clock arithmetic.

OF gravity and electromagnetohydrodynamics and the nuclear forces and yoked pairs.

I found an interesting child's play centre from Holland which is just right for the budding nuclear physiscist.

Equilibrium is the source of all force vectors, and yoked pairs describe the first level of euilibrium action. I suspect there are prime ntuples of yokes!

For a while i will think about my response to shunaya.

Instinctively i sense its magnitude (the Logos Response) its omnipresence (internal to external continuum) its transparency (optical attributes like opacity) its relativity(to me) and its penetrability, its enfolding enclosing lightness that does not smother but wisps of sensations tremble over me through its unrelenting touch.

I have a sensation: i have a thought.

I have an urge to measure, to experience the magnitude and to apprehend it, I have an instinct to organise my sensations: to make sense.

I have an urge to respond, to react and to act and to enact and to re-enact, to rehearse and review what i can recall of my sensations and to remember them to re-engage with them and to manipulate them as tools to remind me , to relate me  and to distinguish me from the void, from shunaya.

My fundamental measuring tool is me.

My Logos Response provides me with regions and boundaries and ratios and proportions. My response is to react and call out, to name each sensation by a response and to remember each response as a name and by this utterly instinctive iterative process to develop an internal language that models my experiential continuum.

And with my language tool i iteratively respond in convoutions to the void developing a mesh of networked ontological connections
Modeling as in a mirror the biological mesh of which i am slowly becoming aware, and ultimately the QCD mesh of motions of which i am gradually becoming aware.

With my language tool i measure and compare and distinguish and record through my biological mesh network and communicate within myself and without. I becomes we and me becomes us.

We come to know.



Of the myriad of proportions that we sense we pick as we may any as our unit for any particular urge that we have.
And so to measure shunaya:

To measure shunaya is a mystical activity, a daily meditation, an intense need that we have to experience and apprehend shunaya. What tools we use become the basis of our science, our iterative convoluted exploration into knowing shunaya.

What tools we use what tools we construct provide no greater knowledge of shunaya, but rather provide us with a different aspect of the workings of our modeling of shunaya. Shunaya is only apprehendable by us as ourselves.

Shunaya is the great mirror in which we see dimly reflections of our own selves.

There are many things in our world, and we learn hat if we pick the right unit we may measure them all. By this experience we learn the value of scale and the operations of scalars, and the peration of scalars is to aggregate and dis-aggregate: we must measure!

The art of measuring is to enjoy the iteration, and so our art is an expression of our iterative urge to measure.

We aggregate to measure and we number: pronunce the names of each aggregate stage in an eternal hymn; we make a count.
We aggregate in bundles to add rhythms to our chant, and our bundles multiply, and so we measure to the sum.
We aggregate in bundles of bundles and so we manipulate adding pattern and complex rhythms to our iteration,factorising our operations and so we sum, and dance and sing and make marks in the sand.

We drag a marker on some surface and this becomes another unit which we manipulate and aggregate and notice how they spin and move and clump together, and cover a surface and so we geometrise, all the while singing and dancing and drawing and building, putting unit blocks together, and learning through our science, through our art, through our play, our singing and dancing  the wonders of shunaya, the applicability of our model of measuring shunaya, our place in shunaya, and what we are relative to shunaya.

The songs and rhythms of aggregation and their counterparts dis-aggregation are the songs of scalars, and they apply universally, but the patterns of aggregation, the rhythms of aggregation are endless and iterative and convoluted and beauiful to us. That one particular rhythm, the indian p-adic rhythm should be so beloved is only a stage in our exploration of the measuring of Shunaya.

We have transcendental rhythms now, rhythms of π and e and yoked pair rhythms of i, and radial rhythms and the rhythms of trochoids, and the rhythms of permutations and combinations, and all of these are the rhythms of our response to our interation with shunaya through unity.

We know. And now we know that we know.....

It is hard to think of a spiral reference frame when you are schooled in cartesian and polar. Yet few of us have seen the full panoply of the polar reference frame!

However this to me is a great model of such a spiral reference frame.

(http://nocache-nocookies.digitalgott.com/gallery/4/492_16_12_10_10_37_43.jpeg)
http://www.fractalforums.com/index.php?action=gallery;sa=view;id=4775

I have moved deeper since those early days of searching and now look for a trochoidal reference frame. Look closer and you will see...

The yoked pairs and the trochoids that form them.

When facing shunaya one faces on the inside a mirror that suddenly shatters one into the myriad of monads that one is.

En. Wahed. Eka. Eureka!

'h¶h echad

Halleluyah!

Shunaya.

"I swear," said the sphere
"That my space is round!
But though i look
In every nook
This truth cannot be found!"

"I see your problem", said the cube
"And it, i think, is due
Perhaps an artefact, i say
Of a certain point of view!
Should you propose
To decompose
I think you will find its true."

Wear your cone!
You silly sphere,
Just like a dunce's hat.
To think that You could find what's round, like squareness
Just like that?

The roots of unity (http://ndp.jct.ac.il/tutorials/complex/node9.html) in the plane sum ro zero, thwerfore the roots of unity are yoked as one would expect in interesting ways that create equilibria both static and dynamic.

One learns from meditating on the odd couple shunaya and monad the structure of my model of the set notFS which I have called FS
And we propose it not as fact but as some concept more fundamental than that

That shunaya emits the roots of unit measure but not as some have taught. The rectilinear and the round cannot be found in shunaya except as members of a dynamic trochoidal class of rotating and rotational symmetries.
The spaciometric version of the roots of unity I will research by and by, but this we know
The yoke of roots of unity are their dynamic rotational symmetry, so that we might say the fluctuating void has rhythm, and swirling in it are rotations of a root of unity. But the roots appear and disappear through the resonance of their yoke, or rather stand as entities of a standing rotational wave.

Therefore the unity of our choice comes with it's own resonance and scalar aggregational rules, it's own combinatorial structure and it's permitted permutability. And it's ceaseless rotational frequency.

The rotation is thus quantised but as we know there is no basis to unity, thus we quantize it by our own sensors that measure, by our own tools that we use to measure.

In this way we feedback loop to ourselves and our unit limitations. The fluctuations of the void appear from our iterative convolutions in processing our signal response to shunaya. The roots of unity inform us that this fluctuation is rotational and trochoidal, combining radial and rotational, indeed defining radial and rotational.

Therefore we imagine the motions of bodies in space, and space itself to arise from these rotational symmetries, like traffic jams and synchronous swimmers, to appear from the relative synchronicity of yoked roots of unity, and to move in coordinated fashion as if a wave, switching on what comes before and switching off what falls behind: a kind of animation through pixel manipulation or rather spacel or voxel manipulation.

And so our prime elemental substance lies in the relative rotational motion of regions of space even below the Planck length, these relativities being perceptible to us through our iterative and convoluted perception processing as attributes to space.

All motion and all stasis relies on these fundamental rotational symmetries which are the roots of unity dynamically manifesting to our sensors in a vast computational flux.

"This", as the saying goes.." this is how we roll!"

These are the roots of unity in the plane, but no longer the Cartesian plane, and yet still the Cartesian measure.

There is no plane, but a space and a tool to measure it with, the Cartesian measure, the Bombelli vector, the Pythagorean unit square diagonally split.

With such a tool we can set up our unit with which to measure a space that is flat and surface like.

 And our units are orthogonal lengths 1,1; √2 a new surd unit at pi/4; a triangle area unit =  half a square unit. These are the units on our measure, and in addition we can do neusis with our measure, so everything is relative. Relative to our measure and contrariwise relative to the form we are measuring, and in addition relative to we who measure. This is the state of affairs as described relative to an observer outside of we who measure, with a measuring tool, a form parametric to us that we have dimensioned from shunaya, a tool of units to be employed iteratively in some grand fractalian scheme of scalars.

And so to notation to record and to display, to calculate and to verify. These four necessary things succinctly put in a form that reveals and does not mystify...how is it to be done?  What praxis shall we use? What mathesis?

In the end we draw upon Descartes and bend him to our needs, in hopes that all familiar with him may follow our line of playful thought.

So now we are nearly ready. On my measuring tool I have no measure for rotation, but I will add one and call it radian and give it a unit of direction called i and a parameter of rotation from that direction called and now we are done.

Rereading Kujonai i realise that he proposed a tetrahedral tiling, not just a triangular one. I think this has been done in the tetrahedral folding of the mandelbrot thread.

In any case i reread Kujonai in the light of yoked decompositions of Shunaya.

Why i not y?
Not why? but y.
Way aye, mon,
Aye!
I eye i
Sign, sigh!
O Shunaya?
Hi!


more word origin (http://www.pballew.net/arithme8.html)
At Jeff Miller's web site on the first use of math symbols I found the following discussion on the origin of the symbol "i" for the square root of negative one:
i for the imaginary unit was first used by Leonhard Euler (1707-1783) in a memoir presented in 1777 but not published until 1794 in his "Institutes of the integral calculus."
On May 5, 1777, Euler addressed to the 'Academiae' the paper  To The 'Academy' the paper "On  Angular differentials  by a formula which nevertheless  mostly results in purely irrational  logarithms, and  integrates the circular arcs  pertaining thereto," which was published posthumously in his "Institutionum calculi integralis," second ed., vol. 4, pp. 183-194, Impensis Academiae Imperialis Scientiarum, Petropoli, 1794.
 
   "For me at least, there is yet another way,  not obvious  to guarantee that this is [it...the correct way], except by proceeding through the imaginary [quantity] , the notational form  which I will designate as i hereafter, so that it may be ii = -1  and thus  1 / i =-i. "
According to Cajori, the next appearance of i in print is by Gauss in 1801 in the Disquisitiones Arithmeticae. Carl Boyer believes that Gauss' adoption of i made it the standard. By 1821, when Cauchy published Cours d'Analyse, the use of i was rather standard, and Cauchy defines i as "as if was a real quantity whose square is equal to -1."
Throughout his Introductio, Euler consistently writes ," denoting by i the infinitely large number of" . Nonetheless, there are a very few occasions where Euler chose i with a different meaning. Thus, chapter XXI (volume 2) of Euler's Introductio contains the first appearance of i as quantitas imaginaria:
For when  negative numbers are logarithmed, results are imaginary  ....... log (-n) will be  the quantity  imaginary, which is equal to i. The citation above is from "Introductio in analysin infinitorum," Lausannae, Apud Marcum-Michaelem Bousquet & socios, M.DCC.XLVIII (1748). Please note that, in this fascinating passage about logarithms, Euler does not introduce the symbol i such that i^2 = -1.
[This entry was contributed by Julio Gonz�lez Cabill�n.]

And translated using google translate to help.

Here we have the terms number and quantity used differently and i think after the greek. Quantity is magnitude of a geometrical form and number is a scalar of unity.

To the greeks Arithmoi were dynamic geometrical forms that scaled and transformed and moved and spread and covered as units. Thus by plethora the units measured as arithmoi, scalars. The idea of number in other cultures was less dynamic, more staid,and scalars were looked on as and for results, Quantity was the same as magnitude, but not as dynamic as greek magnitude, they did not multiply as greek magnitudes do by motion in orthogonal directions by certain lengths. it was more of a rote learning of number bonds.

Cuisenaire rods (http://www.kembo.com.au/cuisenaire.htm) illustrate greek arithmoi.

So negative scalars in logarithms Euler found could be treated as quantities, arithmoi.but only as imaginary ones following rules, and related to log of trig functions.

Euler did not need Cartesian coordinates because he had trig ratios and triangles as his measure, and by relating to De Moivre and a unit circle he was able to tie all 3 together in a way that he says is not obvious, or guaranteed to be correct.He used differentials of De Moivre theorem to obtain infinite Taylor expansions
Cotes had done something similar through studying the log of trig functions nearly a half a century earlier.

Euler treated i tentatively as a unit quantity and therefore applied scalar math to it under the given rules of Bombelli, but as he says in his own notational form, with an innovation of division.

By cauchy it was treated as a real quantity, and by Wessel, Argand (http://en.wikipedia.org/wiki/Complex_number) and De Moivre  (http://www-gap.dcs.st-and.ac.uk/~history/Biographies/De_Moivre.html)  it was given a richer and geometric interpretation. Gauss popularized it , and surveyors and scientists latched onto the geometric cartesian form,

The cartesian measure was recognised as a model for whatever measurement tools were required, thus a measurement tool for the adjugate form was made by Wessel and Wallis and Argand. in which y was replaced by i but associated with y

This tangled mess no one seemed able to satisfactorily sort out, and so piecemeal ad hoc rules grew up into the field as it stands today.

is a real quantity obtained by multiplying a quantity by a quantity, a magnitude by a magnitude but under a set of algorithmic rules. However mathematicians have never stopped teaching students that they are imaginary and complex and they are numbers.

It is time to stop. They are not imaginary,  and they are not numbers, but i is   complex. They are magnitudes and they are units of measurement and they exist in spaciometry as real dynamic aggregations.

When one has looked at the struggle of scientists and artists over the ages to measure shunaya it  is a blessed release to see all their work come to fruition in some elegant and useful form. It is a thrill to look back over the tortured mutterings of feverish minds and gay abandon of playful minds, to look a chancres and dancers, and singers and scientists, at astrologers and philosophers, at artisans and housewives at those in plenty and those in penury mendicant and monastic, egregious and generous,in short all and any human engaged in human survival, and see that they all have contributed to the formulation of the notion of the roots of unity as a measuring tool.

Thus it comes to this.

     is rate of rotation quantity x radian arc length quantity

And all are unit and scalars of units and yoked decompositions of shunaya into roots of unity based on the unit sphere, the dynamic unit sphere,moving relative to any other spatial form.

We need one element for this tool and that is a equilibrium reference from which we can construct all other equilibrium references of relative motion. This I have called a pole as in a pole or a pole star or an pole of orientation. This is a particular radial  of equilibrium in the measure or in the form being relatively measured.
 
The rate of rotation quantity is the root of unity quantity and governs the rotational symmetries of the dynamic sphere; in regards of unit: is to measure dynamic equilibrium rotational states. As with all unities  it is a set of nested yoked unities, but scalar ratios do not apply, these are roots of unity and so by definition are not scalable.

They can however carry a radial scalar that scales only radially, thus providing axial extensions and axial motion and taking orientation from the root of unity.

They also carry a special scalar of sorts called a radian, and this serves to transform one root of unity into another. The radian scalar is motion scalar dividing the rate of motion into smaller unities that relate by rotational symmetry fractal scales to the standard rotational magnitude.

De Moivre  formula gives us the roots of unity in terms of i  because all roots we may choose can ve written in terms of each other, because they are roots of unity.

So defining √-1 as +And- 1 and denoting it by i and laying out clear rules for it's usage we come to the quintessential algebra of spaciometry, to the realm of Wessel and Hamilton, the realm of quaternion algebras..

Many have gone on beyond this, but as pioneers not knowing where or how they would be going, not knowing that they were dissolving into the mists of shunaya.

In the light of spaciometry being the ground of algebra, I look at the theory of operators in algebra.

Really an operator has to be an activity that transforms.

Now the Greeks first created or distinguished unities. Is this an activity that transforms? Here we have to acknowledge the impact of the measurer the uncertainty principle. The creator of the unity the measuring tool defines the magnitudes, dynamic and in equilibrium, that can be measured

Having created the unit we use it or observe it in dynamic action. Is this an activity that transforms?
Again the observer may have transformation that is internal due to an internal act of imagination, or an object may be physically transformed by motions of measuring, applying the unit directly to the form. Or I may even be observed that the form transforms by shattering or dissolution into the unit forms in and of itself. The form may be dynamic alive and active transforming itself by units motions.

Our response to this plethora or aggregation and dis aggregation is to measure, count or evaluate in some way as precisely as we need. We may just compare, we may enumerate units and compare, we may measure and construct units and compare, our response is varied and commensurate to our need.

Thus to abstract to notation of -/+* is to just begin, just to note an observed activity that is transforming. And these notes must be qualified and explained fully in the form of an algorithm based on a set of rules and definitions to wit an algebra.

None of these notations therefore have meaning beyond the meaning given in the algebra. And because of this by design or default algebraists can make their notations confusing and false indicators or clear and customary sign posts. They can invent new notation with the risk of not engaging the usual suspects or twist usual notation to a different purpose, thus deceiving while apparently informing!

Our notation and customary signs and operators have evolved over time to have a flexible and connective meaning that communicates, but also confines and obscures.

The differential calculii and integral calculii are proportions bent to another purpose and so obscured by notation that it strikes fear into the hearts of those who geometrically proportion everyday of their lives!

So our spaciometric activities and observations  give rise to our algebras within which we define our algorithmic operations of measuring, and prior to this we transform our minds by distinguishing which unities we will use to measure and how they are related and yoked together in relative motion.

The radian arc measure is a mod(2 pi) arithmetic but we can utilise other arc measure in a related way but of course not based on the unit circle. Thus a transformation exists through similar circles to any arc measure we require, based on mod arithmetics.

As simple as it is we have confused ourselves by using the notation with a multiple meaning. So y stands for a direction a value in that direction and a variable value in that direction! Thus  3 y we know by teaching is a scaled value in the direction of y  with the variable value of y being scaled.
3i however is different, and the difference is the additional rules we attach to i . These different rules relate to i being + and- in sign but in reality to i being a 4 th root of unity while other values are a 2 th toot of unity. We are not dealing with the same units, so as we know from basic algebra we do not add apples and pears.

We stick by this rule throughout all or measuring, so we do not add metres and centimetres or pounds and pence or even whole numbers and decimal fractions. We always only aggregate like units and we note next to them other units that we aggregate the result is a string of units across the page. The clever thing the Indians did was to make a pattern of these different units so that using mod arithmetic we could move to the next unit.

It is this pattern that shapes our world and our perception of it, for we see  wheels within wheels and structures within structures related by dynamic iteration and geometric convolutions. We see the structure of our fractal universe and we know that it is an illusion, a playful pattern of our choosing. We have other aggregate systems, and we can create yet others.

Choose your own and enjoy the ride!

Of yoked roots of unity and prime magnitudes.


Of yoked roots of unity and determinants.


Of understanding measuring and relations and why notation describes active relating.



Of logarithms base 0, base 1 base π and base e.


Of the relation of mod() and log.

prosthapahaeresis (http://en.wikipedia.org/wiki/Prosthaphaeresis) inspired calculators of all sorts, particularly astronomers,but more particularly John Napier (http://www.thocp.net/reference/sciences/mathematics/logarithm_hist.htm), Napier was fond of playing and calculating (http://history-computer.com/People/NapiersBio.html),and very fond of trigonometry (http://hubpages.com/hub/john-napier), spherical trigonometry (http://en.wikipedia.org/wiki/Spherical_trigonometry) to be precise through which he learned or rediscovered trig identities (http://en.wikipedia.org/wiki/Angle_addition_formula#Angle_sum_and_difference_identities) and the rules of sign (http://en.wikipedia.org/wiki/Plus-minus_sign). No wonder it appears to modern mathematicians that he was looking at imaginary quantities!

Among other things the notion of a ratio  (http://en.wikipedia.org/wiki/Ratio) being viewed and dealt with as a "number" is introduced through spherical trigonometry.
"  Al-Jayyani (989-1079), an Arabic mathematician in Islamic Iberian Peninsula, wrote what some consider the first treatise on spherical trigonometry, circa 1060, entitled The book of unknown arcs of a sphere,[6] in which spherical trigonometry was brought into its modern form. Al-Jayyani's book "contains formulae for right-angle triangles, the general law of sines and the solution of a spherical triangle by means of the polar triangle". This treatise later had a "strong influence on European mathematics", and his "definition of ratios as numbers" and "method of solving a spherical triangle when all sides are unknown" are likely to have influenced Regiomontanus....."

Every aspect of complex numbers (http://en.wikipedia.org/wiki/Law_of_tangents) and sign rules are found in spherical trigoonometry. We also find Napiers interest  in calculation expressed by Napier's pentagon

".... To solve a geometric problem on the sphere, one dissects the relevant figure into right spherical triangles (i.e.: one of the triangle's corner angles is 90°) because one can then use Napier's pentagon:
 
Napier's Circle shows the relations of parts of a right spherical triangle

Napier's pentagon (also known as Napier's circle) is a mnemonic aid that helps to find all relations between the angles in a right spherical triangle. (http://en.wikipedia.org/wiki/Special_right_triangles)

Write the six angles of the triangle (three vertex angles, three arc angles) in the form of a circle, sticking to the order as they appear in the triangle (i.e.: start with a corner angle, write the arc angle of an attached side next to it, proceed with the next corner angle, etc. and close the circle). Then cross out the 90° corner angle and replace the arc angles non-adjacent to it by their complement to 90° (i.e. replace, say, B by 90° − B). The five numbers that you now have on your paper form Napier's Pentagon (or Napier's Circle). For any choice of three angles, one (the middle angle) will be either adjacent to or opposite the other two angles. Then Napier's Rules hold that the sine of the middle angle is equal to:
the product of the tangents of the adjacent angles
the product of the cosines of the opposite angles

The mnemonic for remembering the trigonometric function to use is that the first vowel of the adjective describing each angle (e.g., i for middle) is the first vowel of the name of the function. As an example, starting with the angle , we can obtain the formula:


Using the identities for complementary angles, this becomes:
cos(B) = tan(a)cot(c) = cos(b)sin(A).

See also the Haversine formula, which relates the lengths of sides and angles in spherical triangles in a numerically stable form for navigation......"

Napier found inspiration for calculation in trigonometry and using prosthaphaeresis. Because he was interested in astronomical calculations he wanted to work out the ratios for a very large hypotenuse right angled  (http://en.wikipedia.org/wiki/Right_triangle) triangle giving him very accurate values for distances calculated thereby. He based his idea on rates of motion, because the idea was dynamic, not static, he was trying to show how the sine varies dynamically with change of angle

" John Napier, the Scottish mathematician, published his discovery of logarithms in 1614. His purpose was to assist in the multiplication of quantities that were then called sines. The whole sine was the value of the side of a right angled triangle with a large hypotenuse, say 10^7 units long. His definition was given in terms of relative rates. The logarithm, therefore, of any sine is a number very neerely expressing the line which increased equally in the meene time whiles the line of the whole sine decreased proportionally into that sine, both motions being equal timed and the beginning equally shift.

In modern terminology, L is the logarithm and X the sine. (In modern notation X would be r sin .)  "


We therefore have a clear and expected link between trigonometry and napierian logarithms which Cotes noted as

Changing the angle changes the sin but of course for a constant hypotenuse of 10^7 this changes the cos and the changes are proportional, but particularly identified by the sin. by inventing radianss Cotes was able to notate these relations succinctly and elegantly.

In Cotes form the i actually acta as an identifier signifying which variables are related and the + acts as an aggregate relation indicating a co-relationship between these variables and cos. In Cotes time despite Bombelli the use of √-1 was fraught with misapprehension, Euler often made mistakes using it, so for Cotes to perceive this relation before Euler is due to a different mentality, a mentality based on trigonometry not infinite series. In that mentality √-1 has the meaning of the sign value of sin as its angle rotated through 2π radians.

Thus as we know Trigonometry is at the heart of all methods of calculation especially roots of unity, logarithms and so called "imaginary" quantities.

It is these roots of unity and their absolute link to logarithms (http://en.wikipedia.org/wiki/Logarithm) of all sorts that i think Muses (http://en.wikipedia.org/wiki/Musean_hypernumber) envisioned as a new order of calculus (http://en.wikipedia.org/wiki/Charles_Musès).

Muses words (http://www.intuition.org/txt/muses.htm). and his numbers (http://en.wikipedia.org/wiki/Musean_hypernumber)

The perspicacity of opacity
Is a property that's rare
But you'll find it
When you look for it
Hanging in the air.

The logarithm, therefore, of any sine is a number very neerely expressing the line which increased equally in the meene time  whiles the line of the whole sine decreased proportionally  into that sine, both motions being equal timed and the beginning equally shift.

Thus a constant increase is linked to a proportional decrease, equal additions linked to proportional divisions.

And what are proportional divisions?

Well proportional in this case means according to a proportion. So which proportion?

the beginning equally shift. means that the line bit divided off the sine is equal to the constant line bit added to the log. Thus if the sine is a the line bit added is b then the portions are a-b and a. The proportion is a-b:a.

The proportion we use is: divide into a units an take out a-b units. When using proportions use a physical rod, it makes it real easy.
Now proportional division is dividing the rod iteratively in this proportion at every iteration. This type of iteration is convoluted, each iteration requires a complex activity of self referencing, but a rod in the hand is better than two algebraic variables in the brain!
So the pattern works out an arithmetic compared with a geometric series:

b            a-b   ={a*(a-b)/a}
2b           ((a-b)/a)^1*(a-b)
3b            ((a-b)/a)^2*(a-b)
4b            ((a-b)/a)^3*(a-b)
5b            ((a-b)/a)^4*(a-b)
     ....................
nb            ((a-b)/a)^n-1*(a-b)

So in this case the line bit scales up as the rod proportions down and the iteration counter  records how many (a-b)/a are multiplied. This is called a power, an exponent, but Napier (http://docs.google.com/viewer?a=v&q=cache:Bki-kSr87S8J:webdocs.cs.ualberta.ca/~smillie/Topics/Topics02.pdf+log+sine+tables&hl=en&pid=bl&srcid=ADGEESgHOL0HAfBc44Zb3zleSo2FgqwFBU50Uytl3-EC82xKmKxWkHQ28DAATg78TqjqwXGONJ9Rg7cN-3hdFwduTtTnycMqPSVWtAy8qwNk4M8s9r4CPuK3H6srPfegUnRLdJyP9GL7&sig=AHIEtbTjRNToiQSmZM3WElBNoc4VZ6uZ5Q) called it a ratio  arithmoi(geometrical object).

i can rewrite the nth logarithm relation

nb           ((a-b)/a)^n*a


which gives  (a-b)/a as the base of the logarithm

Napier chose a= 10^7 and b=1

Napiers Logs are therefore not based on e. What they are however are a logarithm based on unity or as near as dammit, and they linked the product of the sines to an increasing logarithm. what that means is that the roots of unity are related to an infinite log scalar arithmetic in ways i have yet to explore, based on log base 1

Thinking about it more this particular log is log based sinπ/2, so this is not the logarithm of sines but the logarithms based sinπ/2.


This leads nicely into the logarithms based cosπ/2 which of course are logarithms based 0
          

Of sines cosines and logarithms and spherical trigonometry in the measuring of shunaya by scientists of all ages. The secrets of the   Yoked roots of unity and how Napier played in the dynamic space of shunaya.

What shall we be riven with?
What nails shall we drive into our souls?
To pin us to the masts of ships
To fix us to their poles?
That commerce may continue thus
Unabated by the storm?
Nay!
That sailing through the universe
We may calculate it's form!

Only a passionate scotsman (http://docs.google.com/viewer?a=v&q=cache:Bki-kSr87S8J:webdocs.cs.ualberta.ca/~smillie/Topics/Topics02.pdf+log+sine+tables&hl=en&pid=bl&srcid=ADGEESgHOL0HAfBc44Zb3zleSo2FgqwFBU50Uytl3-EC82xKmKxWkHQ28DAATg78TqjqwXGONJ9Rg7cN-3hdFwduTtTnycMqPSVWtAy8qwNk4M8s9r4CPuK3H6srPfegUnRLdJyP9GL7&sig=AHIEtbTjRNToiQSmZM3WElBNoc4VZ6uZ5Q) in love with his god would think to waggle a huge Bombelli vector in space just to calculate the sine for the benefit of his honourable astronomer friends! And only an intense irishman (http://wapedia.mobi/en/Classical_Hamiltonian_quaternions?p=1) would follow his lights and make a 3d Bombelli vector that reaches not only outer space but inner space!

Warning, this post is very difficult to follow, and i wrote it? :embarrass:

I will try to improve it, but as a starting point keep in mind the metric and decimal systems! :dink:

The roots of unity are a special term for power relations between trig ratios in the unit circle. As such their link to unity is not as direct roots such that r^n =1 for a given r where r is a "number".

It seems like we have to stretch our concept of number yet again to accomodate these ideas. In point of fact we should not stretch it but rather excoriate it!

 r has to represent an aggregate structure of differing but related unities, which means that one chooses several unities for measuring different scales, aggregates them into a structure, and then treats the aggregate structure  as an entity.

Then  by applying the aggregation rules of that entity for  measuring either motion in space or iterative increase or decrease,whether arithmetic or geometric in formulation, one arrives at the  notion of scalars of 1, which ought to be the overarching unity, that is: the notion of unity that defines the structure of all unities.

As seen above this is a logarithmic process and relates the aggregate entities to the "unity of the logarithm", that is the base of the logarithm. The product of the entities become related to the "zero sum" (of the logarithms of the entities),the modular "zero sum"  of that specified base.

So for example base2 has a modular zero sum of 2 because 2+2=0(mod(2)), etc.

The "unity of the logarithm" is  comparison of a complete or whole or unit clock cycle with unity, In the case of a p-adic logarithmic series each term in the series subsumes all the previous terms, and therefore acts as a whole for the serie thus far. The structure is a nested one, smaller bases/unites being nested within larger ones.

Thus we have a formal relation between the bases in a logarithmic series and the notion of nested unities in a p-adic aggregation structure.

What i mean is if i take the  "roots of unity"  as logarithms, using as  the base the relation( De Moivre's relation, i think)    then i should find that all the logs sum to 0 ( the usual mod(n) completion signal) giving the evaluation (from the definition: the base raised to the power 0 is 1) that the roots of unity are indeed logarithms of the base!

This is very circular to write but this is how i saw it. The only way to write it clearer is to acknowledge this is a recursive or implicit definition defining the roots of unity as the logarithms of the base which sum to zero.

Or to put it another way, as the roots of unity sum to zero they are precisely the logarithms we need to raise the base to , to efine those factors whose product together  is 1

As a concrete illustration: if i use 1 meter and 1 decimeter and form the aggregate 1+(plus and minus)d where d simply is 1 decimeter, the (plus and minus) attribute controls when the decimeter can be added or subtracted from another decimeter during a multiplication of the entity. With the right scalar in front of the decimeter i can get it to disappear or scale up to a meter so as to make the meter disappear, during a set of multiplications.

This kind of control of summation within aggregation by means of an attribute is what "complex" calculation is all about and is not obviously logarithmic,but is "modulo arithmetic" and gate controlled by the +and- scalar combination. This gate control is computer like,storage like and arithmetic logic unit like. It is only when the gate is opened allowing the decimeter value to combine with the meter value , leaving a remainder, that the logarithmic nature comes to the fore. If as Napier did we use really large units we can see then that the logarithmic action is in fact always acting and present. This connotes to using real valued logarithms as opposed to integer valued logs to capture what is happening.

In short, whatever the unities aggregated the +and- attribute provides a signal gate control over the summation, providing the unities are nested, that is scaled ratios. We still cannot combine apples and pears! Logarithms in this case can be constructed to tabularise any multiplication of entities ( aggregations) within the scheme speeding up calculation and enhancing reliability.

Logarithms as an analytical tool serve to systematise calculation and reveal proportional relationships of a geometrical nature.

Roots of unity therefore are not numbers but algorithmic entities controlling calculation logarithmically, and in a way linked to modulo arithmetic of the underlying base. These algorithmic entities are fundamentally based on ratios of the right angled triangle and are "cobbled" together by sign rules and the +and- gate rule, the Bombelli operator.

The operator +, or the notation + is therefore not a simple or even a binary operator but a "control relation" with certain complex relation rules applicable to given identified and identifiable situations. This complex activity therefore distinguishes  + as an algorithm not just a "simple operator" if such a thing exists .

Here is an illustration to meditate on.
(http://nocache-nocookies.digitalgott.com/gallery/5/410_24_02_11_11_17_02.png)

Aha ! Classical Hamiltonian Quaternions! (http://en.wikipedia.org/wiki/Classical_Hamiltonian_quaternions) Nice!

Of Logarithms and modulo/ clock arithmetics.

Aggregation  is usually notated with a + symbol, but this activity is far from simple and far from binary. It is algorithmic in nature and is different to measurement.

Measurement is an activity of apprehension, of response to the environment, of applying or observing the application of unity to a form in plethora, a sensing of the structure and construction of a form by application of the unit measure, or by watching how the unit measure populates a form dynamically.

Aggregation is an activity that we perceive as happening to all sorts of things as they gather or are gathered together.My perception links, yokes, ties things together in a subjective collection. I may even perceive or draw or define  boundary or an envelope for the aggregation that dynamically changes.

Dynamic as these forms structures and aggregates are i still perceive a static form as a resolution, or a completion, a whole an answer a sum etc.

y perception of aggregation and measurement is therefore dynamic and i only "know" when i have an answer when i stop, or it stops being dynamic and becomes stable or equilibrium.

The dynamism allows for rhythm and motion and dance and song and metre and ritual and repetition, iteration and convolution and poetry and rhyme of drawing and painting of surfaces. Sculpture arises out of disaggregation.

So when we look at the Indian decimal system for noting numbers we first have to recognise that it is an aggregation, and then a special rhythmical form of aggregation. It is an aggregation of nested unities, with a gate + algorithm. The gatre+ algorithm controls the addition of the unities on a modulo basis. I could write the gate + as  +modulo(10), meaning only add when this bracket = 10. There is one other thing and that is the power of the modulo, which is its place value power.

So its complete form is +modulo(10^n) where n is 0,1,2,3,4...

I can write the indian rhythm as


   ..........+modulo(10^3)+modulo(10^2) +modulo(10^1)+modulo(10^0)

which i can write as

........a*10^3+b*10^2+c*10^1+d*10^0 wnere ab,c,d are in modulo(10) or mod(9).

There are several ways of notating Napiers Logarithms but this one  makes clear the previous discussions link to them and all calculation

n    =     a*log[((a-1)/a)^n ] where (a-1)/a is the base of the logarithm which is n/a. a is a factor to make the log into an integer.

As previously discussed the logarithm applies to all the values in between the integer value of the log. Therefore each base is portioned  by the log method on a modulo arithmetic scheme: each whole log producing a proportional decrease by the base, so each partial log producing a proportional decrease somewhere in between each base.

Thus  mod(base)*base^ logarithm is a way of representing the naperian notion of antilogarithm as Brigg expressed them in base 10.

Thus Brigg and Napier are really linking the indian rhythm to the dynamic motion of a hypotenuse of a right triangle in the sky, a hypotenuse attached to some moving planet or star, and thus linking its motion to the logarithms base sinπ/2.

Now the indians had already shown themselves masters of calculating sines for their astronomical use so it would appear that the sines were of great use and significance to astronomers of all cultures for calculation, but it was Napier who through dint of hard calculation turned them into a labour saving tool of great utility.

Prosthaparesis thus recommended the trig ratios as a useful source of exploration for quick reliable calculations, and Just Burgi made great use of it and some other methods which turned out to be correspondence relations between series not logarithms (http://hal.inria.fr/inria-00543936)

We find that logarithms are a fundamental form of all computational structures since the indian rhythm was adopted.Logarithms are the basis of polynomials, the basis of our p-adic number system, the analysis of geometric and arithmetic series, the fundamental structure of our aggregation schemes, the link between the entities called roots of unity and the yoked units of shunaya, the essential relation between scale and proportion, an essential link in describing motion and relative motion and a describer of growth and gravitational constraints, and definitely more.

Of measure and of measuring, arithmoi and arithmetic,numero and numbering calculus and calculating , count and accounting reckon and reckoning proportion and proportioning, manipule and manipulating, putative and computation, ratio rate and rationing, gnomon and trigonometry (http://nrich.maths.org/6843&part=) ,vehicle and vector, magnification and magnitude .

We are in a dynamic space we sense it therefore we are. We have sensations therefore we are conscious. We manipulate sensations therefore we think,compute, and consciously measure and manipulate and language. We respond to shunaya and grow in shunaya because we are of shunaya. Our survival imperative our evolutionary imperative are manipulations and computations within shunaya.

Thus we are, and thus we cease to be absumed into shunayas ceaseless computation  and our thinkig consumed by it all our distinct days.

A vector definition as per Hamilton is inadequate, and betrays a misaaprehension of greek apprehensio of the space in which they geometrised. Shunaya has always been dynamic and pregnant for the Hindu, the Buddhist the Confucianist. The greeks branched out into something different, they looked for unity an atom to match the sanskrit "atem". a "non divisible" to match the "spark of self".

All the while their units were in constant motion, like atoms, becoming plethora, being aggregated, disaggregating and dividing and dissolving.

Thus to the greeks their units were dynamic magnitudes.

    A unit is a dynamic magnitude

They watched the units aggregating or being aggregated; or how they aggregated and manipulated them themselves. From this study they learned that motion and dynamism of a unit can and is revealed by using smaller units. As the units decrease in magnitude the detail of motion and structure and relation and relative internal motion as well as external_to_the_unit motion and dynamism of the unit is revealed. This is like magnification of detail, and the greater the magnification the smaller the unit size being used to measure.

The units measure magnitude, relative motion, rotation, twist and stretch and skew, internal relative motion and structure and plethora interaction.

Thus a unit is a dynamic magnitude used to measure and through a measure to display and describe relation, motion, etc.

The measurement tool is therefore fundamental to measuring, describing, and displaying and verifying our exploration of shunaya. Such a measuring tool aggregates units and relations and motions into a handy tools and holds in a static form a representation of a dynamic magnitude or representation of structure of dynamic magnitudes.

When i look at a tool i almost immediately discount the activity i engage in to use the tool. This activity is as important as the tool as it restores dynamism to a static form or structure of our conceived units.

When looking at aggregation rhythms and the +gates of various algorithmic structures, i have looked at a fundamental logarithmic structure.  The logarithmic structure using various bases ties together nearly all rhythmic aggregations.

However their are aggregation rhythms tha are not simply logarithmi:  n! means n*n-1*....*3*2*1.

So an aggregation rhythm based on n! would look like

                            n!+n-1!+*n-2!........+6!+5!+4!+3!+2! +1!

These types of aggregations include
1/1!+1/2!+1/3!+1/4!+......1/n-1!+1/n!+

There are other aggregate based around the permutation and combination formulae. (http://en.wikipedia.org/wiki/Factorial)

I think this aggregarion can be written in terms of the gamma function (http://en.wikipedia.org/wiki/Gamma_function) and their is an analytical treatment of function relations  i do not understand called Verblen functions (http://en.wikipedia.org/wiki/Veblen_function) which may be related.


The Γ function

The function Γ enumerates the ordinals α such that φα(0) = α. Γ0 is the Feferman-Schutte ordinal, i.e. it is the smallest α such that φα(0) = α.

For Γ0 =Γ(1)
 Γβ+1= Γ(β+2)

 the aggregate would be
 
  Γβ+1+ Γβ+.......+ Γ0

where the +gate would be +mod(Γβ+1/Γβ).

I am not certain my notation is correct, but i think the general idea is clear.

David Attenborough illustrated very simply and powerfully that the size of the mammalian brain correlates with social group size, not intelligence.

This is of importance because it gives a diagnostic indicator of autism.

On reflection one might conclude that all whose brain size is below a norm for a social group will be relatively autistic within that group. Thus individuality would enable a new group dynamic to form gradually through evolution of  a family group structure . Such groups would display distinct characteristics called tribal traits, which in fact we could rename relative to a larger societal group as autistic traits.

In addition we would suspect brain dysfunction within a normal brain size that displayed autistic traits.

We know that sexual evolutionary pressure is a major diversifier of group characteristics, and it is of course entirely reasonable to expect brain size to similarly diversify a social population into gangs, tribes , clubs etc, with social order within such groups correlating to brain size within the group.

Now the male-female division has been under investigation for some time and a result has been announced (http://www.sciencedaily.com/releases/2010/12/101222121513.htm) making it clear that the brain has a functional role in sex differentiation over evolution. This differentiation of course is statistical and not genetically set in stone. Therefore the operation of 2 genes are in evidence and the variation through this combination is sufficient to explain the characteristics of male-female gender sexual preference and sexuality, and why some are male inside a female phenotype and female in a male phenotype.

Of course these descriptions are subjective, but a combinatorial study should perhaps show the various population percentages expected within each identified possibility. This would help to demonstrate if this gene effect is real or  a figment of imagination.

I have toyed with the idea of using "recipe" for the idea of function, not only to make it more accessible but to alert those who everyday use recipes to the essential mathematical activity of their labours!

Today many are labouring over recipes traditional and modern in order to celebrate Christmas! Merry Christmas to all you mathematicians who are making enjoyable use of your inate mathematical abilities today, especially you mothers, whose formulations and activities will result in a successful christmas day for all your families.

Shalom

I had thought to look at an aggregation rhythm based on the  logarithms to different bases but realised that these make little sense as aggregates because they are not a rhythmical depiction og plethora or aggregation. They are a signal of a stage in a calculation algorithm

The Greek notion of plethora which taken together with unity forms our basic notion of scalar and scale is a useful one. It covers dynamic increase in the units whether by multiplication, scaling the rhythm of the musical bar while counting, or by division of unit size, setting up sub scales of rhythm within a larger initial unit. Thus a plethora is a dynamic description of unit motion whether observed in a living organism with cell division or a mass production or replication of the unit through some specified agency.
dynamic
Replication of the unit is the macro notion of plethora, whereas plethora through a division process of unit size reduction is the micro notion of plethora, but it is always dynamic.

Aggregation is the apprehension of gathering items together, both physically and psychologically, mentally. An aggregation is a more general structure than form, and is an appreciation of gatheredness. It is not as dynamic and self organising as plethora, but of course could be made of many plethora!

The factorial number system, or as I would draw attention to it, the factorial rhythm aggregation has been explored in relation to probability distributions, and a link to logarithmic aggregate rhythms has been shown, but I would observe the underlying structure of any aggregation purporting to use or relate to logarithms would have to be an antilogarithm aggregate scheme, whereby the base of the contiguos unity in the arrangement will increase by one:

+ mod(n)*n^logarithm+ mod(n-1)*(n-1)^logarithm+......+ mod(2)*2^logarithm+ mod(1)*1^logarithm+mod(0)*0^logarithm

The + gate is not the +mod()  gate we have encountered earlier. The factorial rhythm has a +mod(n!/n-1!)
 + gate. Because of this I cannot make an aggregate rhythm and therefore this type of aggregation is different.

Since this represents an aggregation of rhythmical aggregations each strand of the aggregation is a complete system by itself,so there is no need to move to a different strand to change scale, the logarithm does that , so this aggregation is a ranking or quality aggregation,allowing qualities or ranks to be aggregated.

The +gate thus is a quality yoke and does not allow summation except on an arbitrary imputed assignment. The transform between strands is a ratio of logarithms
Logb(1)/logb(b+1)

When that ratio exists between the strand members those members can be summed at the logarithm level and we know that represents a product at the antilog level.
Apart from rank or quality measurement there may be other uses for such an aggregation mesh of rhythms.

The +gate has become an equivalence relation.

If asked to represent in any artform or mixed media the notion of curved motion, how many would use  stereophonic soundscapes, tactile intensity maps, or gustatory intensity maps?

I have mentioned before the fundamental nature of the Logos Response to apprehending notFS, particularly in relation to proprioception maps. However this is only one aspect of the synaesthesia we use in apprehending shunaya, notFS. 

Spin field (http://www.sciencedaily.com/releases/2010/12/101223144034.htm) effect transistor announced, using the effect of a spin helix field.

And i realise thar polar equations make spirals easy and plot in spacetime.us.

Jakob Steiner is my next post.

Jakob Steiner (http://www.novelguide.com/a/discover/ewb_23/ewb_23_08309.html)  gives me hope that the geometrical/spaciometric basis of Algebra is evidently fundamenta (http://mathdl.maa.org/mathDL/46/?pa=content&sa=viewDocument&nodeId=2344&bodyId=2535)l.

Of the mesh of logarithmic aggregates and the dance rhythms of street dancers and Break dancing.

Of the factorial rhythm aggregate and the factorially fractal structure of infinite possibility space, as it condenses   stochastic probability clouds in regional spaces , containing statistically significant propensities to computational rearrangement: some of which condense into dense regions of relativistic motion space.

Of discontinuous attributable space, to which regionalised properties  are attributed subjectively, and which regions are dependent on the measuring instrument adopted subjectively.

of a dynamic density function that regionalises space into condensing and evaporating potentials, and which is dynamically in equilibrium

3d polar coordinate geometry (http://docs.google.com/viewer?a=v&q=cache:muHZEehvdGEJ:www.geog.le.ac.uk/staff/njt9/lecture1.pdf+3d+polar+coordinate+geometry&hl=en&pid=bl&srcid=ADGEEShTmpkaNidZ4e_qXPher_NU1q4_Fm0QVINmq0WBWSqttyX5eBS6Db5sZ5_CmKmx1u9icl-6fewpS2RbrPqhLs1UYMdtIjVbCyTSoikumIlP0nVrkb0-14J3BqPPsDyL0OfqfACh&sig=AHIEtbRwnq8H-Ob2spuAYUQPDryzgn604A) is quite hard to find a very general treatment of it which is not fully wrapped in tensor or vector notation.


Take a unit pole and a variable rod, and attach the rod o the poke so that it is joined in a corner which is free to rotate. This measring device will be the basis of some measurements of regions, and works by pointing the pole at one point on the boundary of the region and moving the rod around the region boundary noting the scalar length of the rod as it moves round and the radian angle measure between the rod and the pole as it moves in this way. I thus record but clearly not the rotation of the polar axis( the pole) as the rod traces round the region boundary. For this purpose i have an orthogonal unit that is attached to the pole, thus the pole becomes a right triangle, a Bombelli vector, and i have a third marker of unit length orthogonal to the pole but free to rotate in a plane orthogonal the pole . This marker is also fixed on a point on the boundary of the region if possible or some point relative to which the region is fixed.

Now i can measure pole's   axial rotation in radians. This is exactly like using a pair of compasses with the region being traced out by the pencil. Thus we record .

We can then write equations that have to distinguish and and therefore these require +gates and the cos and other trig functions.

When we look at the role of the roots of unity we find that they do one thing, they rotate the unit magnitude in space. So + and - are π rotations of the unit magnitude. The functions that control this rotation and its rate and nature are the trig functions. The + and the - are +gate modifiers, like mod() and control how units are aggregated. and what quantity.


for example.

Standing in the middle of the bank holiday sale motions of people , discerning the aggregation rhythms of their motions, the dance of their hurried predilections.
Suddenly the logarithmic rhythm of their dance gradually emerges.

I think of disaggregation and am immediately surprised by the difference in rhythm. The modulo minus gate -mod() starts with a bang! The rule that you only sum units that are the same in any aggregation applies to disaggregation, but the dissolution of a unit into a smaller nested unit is a shattering surprise! One sees a whole unit dissolve before ones eyes into smaller units that disaggregate and disperse in unexpected ways!

Really the imposed order of aggregation lulls one into a false sense of an orderly build up. But the unfamiliarity of disaggregation means that anything is possible!

So A unit kind of dissolves into smaller units enabling smaller units to be disaggregated. Whole units can obviously be moved away as a whole, or roll away in blocks, etc, but smaller units require  unit to dissolve or  shatter into smaller units.

Therefore the shattering of a plate when it drops on the floor, the shearing off of a precipice of ice from a glacier, a disintegration of a piece of wood into shards and splinters are all examples of the logarithmic rhythms of disaggregation.

So now the aggregation of things has a moment of unification of a larger unit which is somehow less dramatic than the shattering of a unit, but in fact it is more interesting than that because we see these unifications in everyday situations. Traffic jams, where a collection of rhythms forms  solid looking block, that dissipates as the rhythms play out. The formation of planets where a cloud of smaller units surrounds and moves with the larger units. Standing observing a crowd one sees clumping and dispersing of groups of people, larger groups forming and splitting , moving and whirling away and together, finally splitting into single units.

The formation of waves in a dynamic ocean as mounds of water surrounded by smaller dissipating ripples, amonst the coagulating mounds.

This is a great cycle of motion in a motion field experiencing aggregation and disaggregation at the same time, leading to a density distribution that is dynamic and polarised around intense density and extreme rarefaction.

Thus it became a syncopated thought that the logarithmic rhythms of aggregation and disaggregation play their part in the dance of cosmogeny.

What we attribute (http://news.bbc.co.uk/today/hi/today/newsid_7608000/7608874.stm) is what we tend to find.

Aggregation (http://www.orbitaldebris.jsc.nasa.gov/photogallery/beehives.html#geo) and disintegration images

Here's a thought.

When meditating on units dissolving into smaller and smaller units do I observe each unit dissolving into nothing or rather into the finest of tilths deservedly called shunaya?

The possibilities are indeed infinitesimal!

It maybe that the factorial aggregation rhythm allows us to deal with the very very large and the very very small in a sensible way, and that the very very small written in this way are Newton's fluxions, his infinitesimal quantities that apparently "vanish".

Shunaya the alpha and the omega of all, and the mother of all unity if we be the father of it. Man is indeed the measure of all his conceptions, and all our children turn out as we do.

I saw a man in a dug out canoe with a rolled out kite dangling a spider's web over the dynamic waters and thereby catching the largest of fish.

It seemed that the strength in the finest of things when combined with man and dynamic space produces miraculous results, and that an open learning brain may "mind" a new way to do a necessary action for survival. That spider's have a thing or two to teach us about what it is possible to catch with an n dimensional web and that Klein surfaces may just relate to spider webs and roll up into a 3d form on contact with a "seed" form.

And so the wisdom of the trochoids in all their spiral forms web together all the roots of unity into the finest, the silkiest of tilths called shunaya.

And there are webs within webs.....

So to create a spiral trigonometry first pick your spiral.

Then pick your radial hypotenuse.

Then record the ratios as your hypotenuse meanders gracefully along the spiral curve.

Such a table of ratios may have some worth, I think.

The notion of an angle is a "planar" notion having a long history of derivation. However we seem to get scared when it comes to generalising the notion into 3d, because of the overwhelming choice we have for a "corner" or "wedge" notion. Such is life, full of infinite choices!
However I suspect that the most useful generalisation of angle would involve "trig circles" on a unit sphere.

I call them trig circles because the radii would of course be trig function values.
The cones and other polyhedral within these arrangements would form a fractal arrangement of decompositions of 3d space, and would link the roots of unity to such a decomposition.

Nice!

We often confuse our measures with "reality". This of course is a common mistake. They are not reality after all, they are only our measures.

So the question is: can we ever know "reality" without our measures?

The answer is: i do not know. I suspect not as we are fundamentally a network of measuring sensors, and does it really, truthfully matter?

In fact  if it is so, the imputation is that the possibilities are not infinite and there is an absolute unchanging "perfection" from which every measure derives its norm.

As i say, i do not know, but i do not think so.

If, however you wish to think in this way, be my guest. I have no argument with you. Nor do we frankly care. Not said or writ in arrogance, or haughty high mindedness, but in expression of present concerns.

When i speak in higher mathematics to my cats
Or hieroglyphics to the greeks
It's not that i pontificate!
But rather i communicate
What facts
Are pertinent to what one seeks!

Let us suppose,with some foundation, that our understanding of the development of mathematical ideas has been refracted through the predilections and national interests of certain individuals who having found a use for a certain praxis or mathesis propose It as an adherent with some historical or situational persuasiveness in eloquent language of influence.

That such ideas and texts being of utility are redacted or written up into courses of study and learning and promulgated through educational opportunities to eager minds of the young. What may we deduce?

Mainly that nothing presented to us as children is as it seems.

When we become adult, providing we have not become crippled in mind by the process we ought to thoroughly take time to review what we have been taught from our mother's knee to our most revered and wise tutors of education, retaining only that which appears to be useful, and supported by the discoverable data of our research. We may then construct if we so wish our own version of how things came to be, with out the imposition of censure or coercion from long dead philosophers or scientists or other sources of influence.

As there is no basis for unity, I say; that is it is my opinion that this is a notion to be ventured , and a rule of thumb i use in my meanderings, there is no basis for a beginning of this or that idea or a preeminence of this or that notion other than that some wag or wit or respected person has declared it so.

So before Descartes the Greeks and indeed the Indians and earlier cultures managed perfectly well without Cartesian coordinate systems. In fact Descartes managed perfectly well without them and would be a little bemused by the extent to which they have been put to use! For Descartes the praxis was the thing, the mathesis of ordering the thinking and empirical process to come by degree to a solution that is built of smaller mor obvious steps. No crazy notion of mathematical proof existed in his time, just a Greek and gentleman's agreement that if it was obviously so then it was so. The burden then was to demonstrate that it was obviously so.

Of course what is obvious to me is not obvious to you thus some time for meditation is necessary to assist communication of a notion. However in a school or a gymnasium, such time was not given by your competitors and so an element of coercion has always existed in the forced education of the young. Being thus brainwashed was par for the course, but left many educated but dead to innovation and insight. So only those peculiar few who escaped this harsh environment or were resilient to it's grinding conformity ever got to play around with the basic notions of their craft or art.

Men of renown could earn good coin by teaching redolent youth their methods of production, their praxis and thus it is of no surprise that students received harsh treatment for their mistakes, a man's reputation being at stake, and thus his economic survival. The economics of renown therefore are as important in deciphering the development of mathematical notions as the enthusiasm of student or adherent, that is in common parlance the fanaticism of fans!

However it is put or whichever way it is glorified the Cartesian coordinate system did not just magically occur to Descartes . The precursor was trigonometry of the Greeks and Indians and Chinese the utilisation of the ratios of the right angled triangle, the use of shadows cast by a gnomon to measure immeasurable distances by proportional equations.

In his praxis Descartes set himself the task of starting from the simplest of components, and for him this appeared to be the measurement of lengths of a right angled triangle. By Pythagoras theorem and trigonometry all manner of things could be measured and calculated.

Descartes praxis was to show how Euclid can be derived from these simpler concepts of the right angled triangle with as few additional tools as possible. Thus as and when needed and only then he added additional tools in his demonstrations. To think that Descartes imagined a great and rambling use for his crtesian coordinates is to impute to much. Descartes barely acknowledged that they could be used in 3 dimensions, confining himself to weightier matters of philosophy and analytical reductionism.

This is not to say that Descartes was a haughty man. Far from it . His work is engaging and inviting and inclusive and left things for students to do and explore., but his insight was limited even though influential. His jibe at Bombelli's " imaginary numbers" failed to kill off the topic but rather created a reference to it. So he would definitely be surprised at the use of his Cartesian coordinates in the " explanation" of these "impossible" quantities.

And so Descartes coordinates are a much trumpeted version of trigonometry of the right angled triangle, an idea well understood by the Greeks and Indians, but apparently available only to classical scholars in the Western world in dribs and drabs and by catholic educational seminaries.

Despite the seats of learning in the east and far east western scientists were restricted by religious strictures to a received notion of the world and it's origin and it's measurement. Thus Descartes praxis was a way to deduce from foundational ideas the knowledge already deduced by the Greeks. But more importantly, it was a means of spreading to a wider western audience the geometry of the Greeks. What Bombelli had to travel to find out in Rome and other places, Descartes presented dished up on a plate using his own method of deduction and demonstration.

Given that Cartesian coordinates are the systematisation of the right angled triangle geometries of the Greeks we may now proceed to a different understanding of differential and integral calculus.

These topics I venture are applied trigonometry.

Thus the analysis of the tangent to a curve is the application of the tangent ratio of a right angled triangle to a curve or line. Because Greek geometry is dynamic this is the application of neusis to the dynamic situation. The limit process as it is now called initially was simply the tangent at a point being the tangent ratio at that point, found easily enough by similar triangles, that is an expansion to a larger triangle with a hypotenuse parallel to the tangent to the curve!. Thus a table of tangents is sufficient to provide the differential to any curve having drawn the tangents to the curve.

The derivation of formulae for differentials appears normal to the topic, but in fact this applies interpolation formulae to known ponta to find out from table the in between results. Thus the logarithm of sines and cosines derived by Napier were of extreme importance in the development of differentials enabling the full exploration of methods of calculating sines and their interpolations and cosines and from them tangents.

Napier's logarithms were also fundamental to the notion of compound interest, a co related notion to differential calculus and the basis of integration.

As the prevailing interest was in calculating area the area under a curve becomes a simple summation of the areas under the right angled triangles to a curve based on the tangents to a curve. This areas is therefore related to right angled triangle area formulae and proportionately to the cosine of the requisite right angled triangle with the given tangent.

In particular the area of the triangle can be shown to be proportional to some  aggregating product of tangent ratio, thus revealing that integration is a inverse procedure to differentiation which forms the tangent ratios by disaggregation( division).

The details are of interest but do not need to detain us here. Suffice it to say that logarithms sines cosines and tangents are intimately related in the calculus and are all applications of the Greek theory of proportions and right angled triangle trigonometry, or rather trigonometry of whatever flavour Greek Indian or Chinese or Arabic.

The link through spherical trigonometry to astronomy and astrology etc has to be mentioned but more importantly the notion of a dynamic magnitude that measures geometrically dynamic situations is an old idea too, which we have disguised once again under the notion of vector, admittedly a notion advanced by Hamilton but not a new one, rather a restating in modern clothes of an old ancient view of the dynamic nature of shunaya.

It occurs to me that my measure with a little modification becomes recogniseable as a surveyors theodolite (http://en.wikipedia.org/wiki/Theodolite). This essentially renders surveyors as potentially supreme spherical coordinate producers..

I wonder how they record their data?

Then there seem to be geodetic measures (http://en.wikipedia.org/wiki/Deformation_monitoring) and ways of measuring the whole earth (http://en.wikipedia.org/wiki/Geodesy)

It seems that geodetic  measures are more like my measure, because surveying seems only to record one angle, but geodesy (http://docs.google.com/viewer?a=v&q=cache:kx7E91uCWF0J:www.lepora.com/examples/Geodesy.pdf+Geodetic+measuring&hl=en&pid=bl&srcid=ADGEESh9FfI4Wj1UlLtcc_pQvVAU-A-Hoe2SHp171yqoa9V4vXBoS1YlEx1EgJk5Sx3MRHBL2c798omhF43t23Clqwn7fvwY-yfg022gTdxlJSjAe-Ki9zlZ2Tw88KBgJ-wlAlbk7BfR&sig=AHIEtbSzvgXAQ-JQy-qhGu8lWXKElcs92w) measures more than angles (http://www.wisegeek.com/what-is-geodesy.htm)

Finally found what i was looking for ! Geodesics for Dummies (http://docs.google.com/viewer?a=v&q=cache:sudmJ8OEqo8J:gge.unb.ca/Resources/GeodesyTutorial.pdf+Geodetic+measuring+for+dummies&hl=en&pid=bl&srcid=ADGEESi8_HP_XfHTOFcC2LfzcqK-H8tP2WlFkN8FPHq27mkjxGSBQLiZOAXrnkvPJhNM75iPsDVl3JZIRzj4z8IHPFi7Gp0Te1_iqgE3rkWu008cSOU5HEIMtjip7JVuvdgFleOy8Y5m&sig=AHIEtbSGu1yd8vkhW6atRmfi9e-bMYvDlA)! Nice!

For symmetrical forms a spiral or vorticular path is the most efficient space filling "tiling" path, as the inherent rottion is disguised by the symmetry.

For all other forms the most efficient path splits into efficient paths of rotated forms, and these too are spiral or vorticular . In fact all paths are trochoidal in nature and relation, trochoids of spirals/vortex. These trochoids support either an inward spiral/vortex. an outward spira/vortexl or a closed circuit such as an ellipse/ ellipsoid. In special cases a circular/spherical closed circuit is observed.

We may derive trigonometric tables for any boundary, thus the special case of circular or spherical rotation is not the only trig database we could develop. Each of these databases would be a fingerprint of a form up to similarity, no matter what orientation, but corrections for parallax and perspective would need to be included.

We can even derive trig tables for open or dynamically varying forms, if they vary systematically by some trochoidal path.

The number concept makes fractions difficult to comprehend. Proportions are the natural way to understand the scalar nature of spatial measure. Units and nity are natural constructs for the human brain, the only difference is the scale sizes of these unities, the ratios and proportions.

Fractions seem alien and are indeed are alien to our natural way of measuring. Our neural networks and meshes work from the smallest common unit signal.

Now the modification to the measure discussed earlier is based on 2 Bombelli vectors. One is a half unit square cut diagonally (π/4 set square) and the other a Bombelli vector with a unit side while the other 2 are variable, but always forming a right angled triangle.

Hinging these 2 vectors at the right angled corner allows free movement in a unit sphere which can be used to determine radian relative angle motion. The semi unit square Bombelli vector i will denote as the pole and the variable Bombelli vector i will denote as the rod, and the angles of rotation will be described by spherical triangles on the unit sphere.

This will be called a quaternion measure.

Units are dynamic magnitudes. There appear to be some basic dynamisms: growing  and shrinking, rotating relative to a pole, radial translation from a centre region, shearing and twisting morphisms relative to a plane, rarefaction and condensing.

For growing , shrinking and radial translation i can use units that "plethorate" that i can scale by replicating and aggregating or disaggregating using +gate algorithms.

For rotating and twisting i need units that "modulate", They reflect the cycle of a rotatin object. This distinctive behaviour requires modul or clock enumeration systems. and aggregation requires +mod() gates

For shear we require units that reflect conservation of area and/or volume but not perimeter or boundary magnitudes. A curious unit which i have yet to explore.

For rarefaction and condensing we need units that conserve kinaesthetic sensations of pressure but again not boundary magnitudes, so proprioception of geometry is different despite kinaesthetic pressure being the same. Sensitivity to this type of unit will depend on sensor density, so that the average over a given number of sensors will be different as the unit boundary magnitudes grow smaller, eventually resulting in spike like signals like Diracs integral for impulse force distributions.

How these last units are aggregated i do not know yet, but it seems clear that some regional limitation will apply, requiring an iterative approach to aggregation, therefore a fractal arrangement of some description is still the outcome. I suspect that nested unities will be involved with some weighted function controlling the +gates, the functions possibly using boundary parameters. Thus for shear the bigger he boundary the less impact on the overall aggregate, whereas for density or rarefaction, the bigger the boundary the greater the impact.

For units of rotation i can distinguish four +gates of renown: +, +&-, - ,-&+. Bombelli's rules apply.
+&- tends to be written a i when it is a rotation, when i is an axis it is the same as the y axis and not a rotation.
-&+ would then be i^3 or equivalently -i.

The confusion between axes and rotations i have worked through in polynomial rotations which is unnecessarily complicated because of it!

When i think about the +mod() gate for rotations i see that though the base is 4 the aggregation is not a logarithmic rhythm. Thus i venture that roots of unity are not aggregations that can be set out in an obvious logarithmic form. They have to be written as cosines and sines to do that, which is where Cotes formula reveals something about rotational +gates: they are in fact yoked roots of unity.

Aggregation algorithms are therefore really interesting operators, and require some tabulation of examples to establish. This is as close as one gets pragmatically to axiomatic algebras: a detailed model of the parameters and their interactions has to be set forth initially before any general arithmetic can be done. Thus Brahmagupta, Napier, Bombelli and Hamilton and numerous others  laid the foundation by detailing these relationships, or bonds.

Some basic sequence definitions

http://vimeo.com/18347514
click here and remove leading '  (http://'http://vimeo.com/18347514)

A sensible explanation of recursive / convolution

http://vimeo.com/18347244
click here and remove leading '  (http://'http://vimeo.com/18347244)


so iteration can be understood as going round the convolution/recursion again with the new value.

In algorithmic terms the recursion/ convolution is a systematic sequence of operations on a start point resulting in an end point or teleos, that is a goal, which is necessary and sufficient for the next iteration of the recursion, where iteration means using this goal result in the recursion as a new start point.

As complicated as it reads most activities in our lives are recursive/convoluted, and we get to where we want to be, or where we end up 9depending on how one perceives it) by iterating the convolutions in our lives r being iterated though th convolutions in our experiences.

Why functions are recipes

http://vimeo.com/18329532
click here and remove leading '  (http://'http://vimeo.com/18329532)

And Lazurus Plath (http://vimeo.com/user524869) gets round to explaining a bit of how his applications work

http://vimeo.com/16337818
click here and remove leading '  (http://'http://vimeo.com/16337818)
Check out his blogspot.

For me Lazarus Plath is a genius. I would rate Terry Gintz as a genius and Kegan. J. Brill too.

Lazarus as yet has been hard to talk to, he has not responded , but I am going to keep trying.

Taking radius and circle as his only forms he makes spin the basic dynamic and then uses relativity of spin centres to create a dynamic magnitude which I will call a radius vector sum.

I have to step back a little bit and describe what I see.

I see a spin field . The spin field is a motion field which rotates around a centre. We think we know what that means and would ordinarily take that as axiomatic.

However I have analysed this notion spaciometrically and determined that in general we attribute spin under the notion of apprehending the boundary of a form. During that process we attribute the notions of centre , axis, and radius. These are static perceptions of dynamic magnitudes.

We also attribute a relation to ourselves which we generalise as relativity.

This does not arise mystically through intuition, as I once thought, but no less mystically: through an innate application of a measuring neural network, which has a teleos which is to compute to an invariant solution.

Thus all the motion attributes are a curious mix of these computational processes and my sensing of notFS or shunaya.

Conceptually I cannot sense outside of this process, nor truly imagine outside of it. Thus my imagination is intrinsically inward an self referencing and self aware.

Despite the immense unending sense of freedom and joy, I cannot perceive outsde of this self referencing cybernetic loop, and any thought that I am or can is in fact a self referencing artefact, an alternative point of view which provides me wit eternal interest and joy.and a sense of completeness.

Thus I attribute stasis and equilibrium to certain perceptions through invariance or as one might suspect "almost invariants" that is a set of constants that are " good enough" to be used as invariant, but which are transcendental or irrational as ratios and which hold our neural processing networks in an endless computational cycle, thus giving the appearance of stability and invariance, while actually gradually, infinitesimally changing.

Contrary to this I attribute dynamism to various forms that have a relative stable structure, called a tensor, which is invariant under a set of transformations, which we call affine. In fact these transforms are computational outputs of the neural network in response to sensing these relativities within a form, but no equilibrium/ invariance between this computed cluster and any other computed stable forms in the background. This we attribute as motion to or of a " body", the computed cluster, and it is inescapably relative.

So my measuring network is able to provide me with innate notions of dynamic magnitudes which are always inherently relative within and without a form.

The notion of a spin field is therefore an abstracted foundational conception in which I would seek to show by some proprioception attribute a connectedness to every notion of dynamic magnitude, and to demonstrate a spaciometry adequate and necessary to encompass all observable phenomena.

So I note through lazarus's work that using just the notions of a circle and the idea of unity which means plethoration and ratio as scalars, and fixed or invariant magnitudes which have other" degrees " of freedom as well as other imposed constraints, I can rigorously define a planar radial" vector sum" .

This is where we have to shake off the notion of sum as being a somehow real entity, and realise it as a conceptually directed relation which directs the attention and focus of perception to the required magnitude.

Thus a vector sum in a plane requires us to look at and measure the diagonals of parallelograms, with contiguous edges of given  magnitudes .

I would rather call these dynamic magnitude measures and specify the measure that needs to be focussed on. To measure requires a tool for measuring, and this spaviometrically will perform the calculation faster than  writing it and rotting it up!.

Finally these dynamic magnitudes are typically volumes from which we can isolate relative areas and relative lengths as abstractions.

The aggregation of these dynamic magnitudes is also a dynamic affair, but of interest here is he dynamic aggregation of magnitudes that are spinning in the spin field or as the spin field.

As the spin field we tend to aggregate spin in revolutions, but we do not count revolutions much, preferring to measure them by a revolution counter, due to the intense boredom that ensues due to the mod() arithmetic!

Of the continuing strangeness of imaginary magnitudes,but unnecessarily so.

So when Brahmagupta invented misfortunate numbers after meditating on the I ching he set in motion a wave or revulsion and confusion that exists till this day.

The greeks had drilled down to the core of magnitude, looking for unity, the indivisible unit of reality they called an atom.
For them, and for all cultures there was always something, never "nothing", and whatever this something was it was dynamic.

So Brahmagupta was traduced in translation! and then he was rejected through his unfortunate choice of examplars: misfortune and fortune!

Shunaya is the finest of tilths, the lightest of mists, the essence of perfume! but it was reduced to The cold blackness of misunderstanding.

Brahmagupa's operator, Bombelli's operator speak of yoked magnitudes, and the yoke that distinguishes them is the yoke of roots of unity, the yoke of rotation, the yoke of dynamic equilibrium.

The greeks , From Thales to Plato with their plethorate arithmoi, and their proportional relationships, developed the notion of scalar. The indians with their love of rhythm and vedic  sutras and verse developed the notion of compact aggregational structures.Both had a dynamic understanding of a motile reality, and responded to it dynamically.

So the concept of numbers has arisen as an amalgam of many notions but non more injurious than the conception first introduced by spherical geometrists and gradually developed in the west into the number line by Wallis and eventually Dedekind. The dynamic magnitude was being turned into a static symbol, a shadow of itself, and keeping all the bad associations without a hope of redeeming all the good dynamic qualities.

The dynamic qualities were spread abroad the landscape as one argument after another rigorously defined a number, leaving out this or that attribute. As the attributes of dynamic magnitudes  were discarded others came along collecting them into new entities called vectors, matrices, tensors and eventually a strange collection of measures were put in place of the dynamic magnitude of the indians and the Greeks.

So none treated dynamic magnitudes properly, and thus  weird notions proliferated and the weirdest was the notion of the "negative number" along with the notion of "nothing".

Brahmagupta understood from the I ching that unities emanated from shunaya in yoked pairs: the yoked pairs  swirled around like spokes in a wheel, in equilibrium, dynamic and or static. They were balanced and yoked and from Shunaya, the womb of all things, thus they were always the same as shunaya when aggregated, and they had their own rhythm.

These yoked pairs were balanced and therefore equal, but moving in a rotation and therefore constantly changing their orientation relative to "the world", but not to themselves. Brahmagupta mistakenly chose to distinguish them by fortune and misfortune, but we do not have to make the same mistake. I will distinguish them by their angular difference in radians 0 and π. or i could use π and 2π.

So now i can use 0 to distinguish all magnitudes which are yoked to π magnitudes.

If i use a boundary i can write these yoked magnitudes on the boundary with either a 0 or a π  and place them diametrically opposite. I could even use _0,π to further distinguish the yoke and let the magnitude and direction of the yoke be represented by a line. All lines cross at an intersection which i will call shunaya, the centre the origin, the balance point.

If as Brahmagupta advises all the magnitudes are unity we have 1₀---1_π

And these aggregated are shunaya, the centre from which they came and around which they spin.


We have to recognise the rhetoric of former times. Thus we did not use a confusion of symbols to discuss these things as they were not in use then. Instead i attempted to describe, distinguish and demonstrate the notions to which i draw your attention, just as ancient scientists did, and consequently their rhetoric was misunderstood and lost in translation.

The Sutras allow us to apply Brahmagupta's advice  in this way and we see that a dynamic magnitude which emanates from shunaya is balance exactly by a dynamic magnitude which emanates precisely π radians rotated from it. Thus shunaya is Equilibrium of all things, also.

Every dynamic magnitude has a radial direction and a relative rotational magnitude to its balance magnitude. In a yoked pair this relative rotational magnitude is π radians, but as we now know there are more relationships than yoked pairs emanating from shunaya as roots of unity!

The important thing is that a dynamic magnitude has a radial direction and a relative rotation associated with it, and the vector notion is thus shown to be inadequate.

This was the conception allowed by Brahmagupta's attribution of yoked pairs to shunaya. Thi is a conception that Brahmagupta placed before us, but though original it is derived from preceding notions abour space, or shunaya from other traditions and cultures.

When Brahmagupta applied it to astronomy, he added additional rotational attributes to the yoked magnitudes, namely radial axis spin in seemingly opposite clockface  rotations. He also constructed a table of sines to help with his astronomical calculations and measurements, which i think like the reek was related to right angled triangles and chords, but additionally to rotating circle radials by Brahmagupta.

Standard trochoid (http://home.comcast.net/~trochoid/WebTro1.html) Use this web app to meditate.

Lazarus is not very communicative, but he is brilliant.

The boxes represent circles that are linked to one another, they are related, relative and yoked.

In the box the top control measures the radius rotation against the grey background. It measures it in revolutions. 0 means no rotation of the radius, 1 is the unit revolution, 2 is two revolutions per unit revolution etc.This measure affects the radius, and the radius is made visible by clicking one of the little squares in the top left hand corner.

The third control in the box, the lowest one is the circle radius measure.The radius of each circle is yoked to the centre of the next.The linking is supposed to be orthogonal, but of course they are actually generalised relationships, a type of Lagrangian set of constraints.

The yoke between the centres ha magnitude the sum of the two radii measures. The top control sets the rate these radii rotate against the background in units of per evolution, the app draws a rotating radius, or a rotating paraleleogram or a rotating hypercuboid  etc.

These dynamic forms are dynamic magnitudes that we otherwise call "vector sums", but this is an unnatural term which is best replaced by the dynamic magnitudes on screen.

The yoked relations are worth meditating on.

The very first new thing is that the proprioceptive mesh has a decision to make: which of all the sensory data does it take as the referential base?

Proprioception "draws" my geometry for me, it puts in place all the relations and connections in my immediately surrounding space, and ties all the other sensory mesh responses to this basic platform. Thus relativity is a choice of reference frame made by my proprioception. Once chosen the data is "snapped" to it to make a "sensible", ordered arrangement of space from that reference frame. As each reference frame is chosen the data is reorganised immediately relative it. Therefore i have to face a choice, a moment  of proprioceptive indecision. The organisation of the sensory data actually is changed by which reference frame i proprioceptively adopt!

I can see a spiral, a straight line, a complex rotating system, all depending on which proprioceptive response i take.

Proprioception works like this: suppose I take a picture of a scene. If I take my picture, on my retinas as my basis I then proprioceptively step into the seen and attribute behaviours and properties to every object in the scene. I do this continuously through an iterative process of adjustment requiring me to compute scene changes and relations on a masive scale every moment. If I change my focus or attention, then proprioception takes that into account and provides valid scenarios for the new focus.

Sometimes the proprioceptively map does not quite fit the sensory data, and that is when I have paradoxes, illusions or misrepresentation of what is in the scene.

I may provide a perception of a scene proprioceptively which is lacking detail, but sufficient for purpose of navigation, situational awareness, echo location etc. But for detailed examination it is insufficient, and that is when I will focus on a small range of not FS a small region of space and enhance the data signal and develop a more detailed perception by iteration of the perceptual output . This new set of perceptions are organised into levels of percption so that I/ we can choose appropriately how much processing time we need to devote to recognising something.

Recognition then is perceiving at various levels of processing. Thus a perceptual process iterates through various levels adding more detail to the recognition and testinfpg the validity. Should the validity test fail recognition becomes confusion and should prompt reexamination at a slower and more detailed rate.

The rate of recognition is dependent on the numerosity of the perceptions and these are determined by the experience of the perceived, the fluidity of the processing state, and the decision/ control power the perceived has at the time. Certainly one can be more decisive after a rest, and with a clear head. So flooding the process with perceptions derived from sensory and proprioceptively meshes will not bring about quick recognition, but it may lead to the correct recognition with sifting of the data/ perceptual outputs or it may lead to new relations andn creative links.

There is a kind of homeostasis in our processing system, best expressed by occurs razor: the simplest recognition is the one we tend to use first. It may not be very good, but if it saves my life it is good enough!

Into this mesh of activity and perceptual behaviour comes the notion of method. Thus we develop individually a method of processing or managing all this computation that is unique to us. This method or collection of methods is then re iterates into the mesh and used in a self referential way to define our self identity.

Thus a particular perception behaviour distinguishes an individual, and the common perceptual behaviours distinguish groups of like minded individuals.

Thus scientists perceive in a particular iterative pattern, mathematicians in a distinct but similar pattern and so on for all the so called areas of study.

At the end of the day each one of us is a unique combination of perceptual traits that focus our thinking and behaviour in ways that are adaptive or maladaptive useful or dysfunctional, depending on the prevailing situation.

As my son is fond of remarking: if ever we get attacked by zombies he knows exactly how to save us all!


So relativity is firmly based on the proprioceptively perceptual mesh I use to apprehend a circumstance, and when I meditate on lazurus Plath's trochoids in motion I see this very clearly and intriguingly.

In Lazarus trochoid browser app, the box contains from top to bottom

Radius rotator control
Radial ratio control
Radius control.


The radial ratio control is the largest figure in the box and it is the ratio of the arm link to the point drawing the trochoid curve to the radius of the circle rotating the arm link from it's centre.

This may make sense to you but not to me, when I look at Lazarus app. What I see is a White point with a grey trochoid path running through it which the White poit always stays on and travels round. But for some settings the point does not travel the trochoid curve does. So what is going on?

Well, so far I think Lazarus has used several output layers and combined them into one screen. Each output layer shows the result of different aspects of the calculation. Thus the trochoid is an out put of a normal java trochoid app; the radii are a drawing app that are timed to sync with the trochoid app, so that for each radial plot a trochoid is drawn going through that White point. The White point geometrically is very straight forward and is always the displacements from the pixel map of the radii. This draws a parallelogram which by the way is an arithmoi and in particular the aritmoi we use to define a vector sum.

So how does Lazarus get the trochoid to stick with the White point no matter what the trochoid is doing- and sometimes it is rotating relative to the background as well as the radii!

And the effect of purporting different aspects to different masks is that it is easier to see the absolutely wonderful relative symmetry of the whole system.

In this app the radial ratio determines the shape of the trochoid but not it's geometry apparently, and that is the strangeness. The definition I started out with does not seem to explain this. If the arm ratio increases should that not lead to a change in sizes/ magnitudes for a given radius? In short smaller or larger circles should be evident, shouldn't they? Or is there an alternative explanation of this ratio in terms of rates of rotation?

Limits and continuity are related through ratio, similarity and magnification, in the following way: Proprioception and other mesh sensory systems compute a spaciometry. The system attempts to compute this efficiently and quickly and uses fast transforms of any changing parameters in a scene.

This is very similar to video encoding algorithms, particularly the VP* encoders by Google/ON2 , a brilliant codec developer swallowed up inside google somewhere!

Thus frame, frame rates, keyframes etc are all appropriate and proportionally applied to the discussion of computing a proprioceptive spaciometry of notFS.

So the limits that we inately expect are iteration or computation limits. This means that the mesh expects to compute a frame within so many iterations. When this is done the surface computed is sufficent for the purpose it was computed, containing detail or artifact def pending on the intentional teleos of the whole system.

 therefore like Albert (http://www.poemhunter.com/poem/albert-and-the-lion/) we may at first see a lion, but then on a second look see a different form all together. (In fact this is the poem (http://poetry.poetryx.com/poems/1137/) i remember a snippet from from my youth! I never knew until today!)

The limit of the perceptual computation is linked to magnification by unit size. thus the unit size allows the same computational limit to compute detail for a smaller area/volume.

As the unit size is gradually shrunk as in magnification the copputation smoothly adjusts, giving the ene of continuity and contiguity. However speed of change soon dispels this illusion, For very fast change we will experience a blur or even no resolved image at all. This shows that the actual event from which the mesh is calculating a response to is changing faster than it can compute , or there is no contiguity and/or continuity at that region of sensory data, or some intense convolution exists there.

Thus at all times our/my notion of limit and continuousness is a computational artifact of my proprioceptive and other sensory meshes.

I can reasonably expect my sensory meshes through the CNS and PNS to compute fractal scenes at any magnification, or unit size as long as the signal input variation is there for them to process.  

Lazaus trochoid java app has at last revealed some answers.

In the top left hand corner above the circle boxes are some small squares which are controls for various settings. The first reveals a text box containing parameters and settings which i do not understand and cannot seem to change anyway. The next bu one reveals the radii and the geometriv magnitudes and forms, the next but one reveals the masks and some other aspects of the app, possibly surface shading? the next but 1 generates instances of circle boxes, that is you can place as many circles on screen as you like.

Now the masks reveal that the first control which i called Radius rotation actually generates a trochoid in black. The second larger control which i called radial ratio actually generates a trochoid in green and the 2 trochoids rotate relative to the background. There are some trochoid meshes related to the green trochoid which vary according to the diagonal measure of dynamic form produced by the radii(vector sum) and the white dot traverses both trochoids simultaneously!

Lazarus chooses he green trochoid to represent on the main screen/mask.

Looking closely at the trochoid mesh there are some unexplained short trochoid paths cutting across.

This still does not explain the arm ratio idea, and in fact it seems to support a geometric explanation of the path of the trochoids bases solely on rotation ratios and the dynamic diagonal magnitude of the geometric form.

There is a curious attribute of rotatio of a form we take in our stride, but nevertheless is very interesting. As an object rotates it defines an axis of rotation. The notion is hardly explainable without a demonstration, and is an example of a recursive definition, a convolution or a tautology!

It seems everything to do with rotation is circular,but this notion needs to be avoided: we are dealing with closed loops or near closed loops of any path, we could include spirals, but we have to recognise the dependence of their definition on radials and closed loops. The circle is only one form of closed loop, and a very special one.

So we have an axis of rotation whnever we rotate that is apprehend a closed loop, and with that notion of axis is an unbreakable link to a radius or a radial line if this line is not constant in length.

So a closed loop, an axis of rotation and a radial line are tautologically linked to any definition of rotation. We simultaneously observe and appreciate measure and compute all 3 when we perceive rotation.

This simultaneity develops an innate sense of synchronicity. This relate Chronos to rotation. Their are other forms of simultaneity but unless they loop back they go out of observable range, therefore closed loop simultaneity is best for Chronos measurement.

Chronos measurement requires a unit as does all measurement. In this case i can use a unit rotation around any closed loop. We rhen reference any closed loop rotations against this.

This of course produces scalar multiples, but leads to part rotation issues, and synchronicity issues. There are also issues with  regard to rotational quickness and constancy of quickness.

We apprehend quickness innately through our fundamental sensory mesh processing cycle frequency. Thus our basic iteration cycle frequency is an inate measure of quickness of change, and therefore all motion, including rotation.

Rotation cannot be measured by smaller closed loops, it gives the same result, and rotation frequency requires a unit rotation to establish, so the only thing left to measure rotationally is part rotations.

Part rotations lead to the notion of arc, not necessarily circular, which leads to the notion of angle between radials.

The precision way to divide angles into smaller and smaller parts has taken centuries to perfect. and is based on the circular arc.

With the angle measure established we can talk about angular magnitude as defining to rotational magnitude

Angular magnitude can e aggregated as i have discussed in the previous posts on aggregation rhythms, but because it relates to a closed loop measurement of a dynamic rotational motion the mod() arithmetic is particularly appopriate.

The Babylonians used a mod(60) clock arithmetic and a mod(60)60ⁿ aggregation structure which include n=-1,-2 for minutes and seconds of arc.Thus there whole aggregation rhythm was tied to this rotational scheme by setting unit rotation=360.
Using mod(360) as a base it was easy to switch between everyday aggregation needs for commerce and astronomical aggregation needs. These astronomical needs related to yearly cycles of the sun, stars and moon and provided a strong astrological link to everyday commerce. and the notion of lucky numbers and good days and bad days on which to venture.

Today we tend to restrict our use of angle measure to a range of (-π,π] but there is no reason why we cannot establish an aggregation rhythm mod(π) and an aggregation structure mod(π)πⁿ and use mod(2π) to factor out the rotations.

If we can write for the terms.

i have concentrated on the rotational and angular magnitude of the rotational motion, but have yet to deal with the synchronicity.

I can use a unit rotation as a frequency measure. One aspect of measuring rotation is to establish a radial as a marker an some reference external to the rotating radial and a counter to record the radial marker lining up with the external reference. This can be counted, enumerated, or experienced as a duration.

Duration is an experience which is proprioceptively induced. The normal mesh clock rhythm for processing data is attached to some simultaneously moving object by a process of reference. Certain proprioceptive responses require severs processing cycles to complete others are instantaneous. The sense of duration is therefore an artifact of the processing processes, which is associated simultaneously with some moving or changing object..

The final strange thing is the circumference of a circle. If as we do we have a radial arrangement of a disk of material and that rotates, it is hard to believe that it will stay together due to the increased circumference regions have to travel to remain in synchronicity.

Lazarus Plath's circle box controls  use rotational frequency ratios to determine the shape of the trochoid. This relates precisely to roots of unity!

This is not very clear, or thought through properly, but i sense that it may be possible to define synchronicity by a lagrangian set of constraints  for a rotating body made up of regions.

 the constraints would take equilibrium as the fundamental notion and the regions would rotate relativistically around  local centres of rotation, as well as around a centre of rotation for the whole collection or regions.

 a rule of contiguity would define the maximum displacement difference between a region and its local rotational centre before which rotational constraints could be applied to determine the local relative position, and after which the rotation around the body centre determines position of the region, or rather what i mean is the position of a region is determined by the dynamic magnitude formed by the body central rotation and the local centre rotation, with a cut off where the local centre no longer affects the position of the region because the body centre rotation creates to big a displacement, At that time the region is no longer conected to the body, as this is achieved through local equilibrium conditions, and therefor the body no longer has a rotational influence on the region.

At this radius the body simply breaks up into separate regions which may clump together hrough local rotational centre effects, independent of the main body.

This is a kind of gravitational system transmitting equilibrium and synchronicity through local rotational centres, with Lagrangian constraints.

I can see it but i have not fully described it.

The question is begged: what happens when  a region "breaks free" from a local rotational centre?  This till leaves the bodies rotational centre as the centre of rotation for the region, but without the constraint of synchronicity. This means that a region in this state is free to rotate around the body centre, but is notsynchronized to the centres rotation, so it has an extra set of degrres of freedom, precisely the "broken" lagrangian constraints.

However why would it rotate around the centre of the body? I cannot answer that as it is a mystical, divination question which is equally answered by: "equilibrium constraints" or "god would have it so!"

Newton chose a combination of the two and called it "action at a distance" and described how it works.

The Equilibrium constraint relates again to Shunaya, yoked roots of unity and rotational ratios, and therefore will be logarithmic in nature. Newton's estimate of the proportions was an inverse square "law"(presumably of god) of the radius yoke, but it is in fact a negative logarithm law of the roots of unity using radius rotation ratios of some form. Depending on the frame of reference the paths of motion of the "broken free" regions would describe trochoids .

OMG! Lazarus has organized the control boxes in the circle box so they use the same radii. The rotational ratios are varied and that defines the trochoids for each of the 2 control boxes, but only one of the trochoids is displayed on the main screen and only one of the rotational radii frequencies is shown or picked out; the common one between the circle boxes not within the circle box. However the relative motion between the two trochoids is displayed, so the main trochoid rotates against the background, but by a function of its relative position to the undisplayed one.

Then Lazarus does something amazing. When a third circle is brought into play he calculates the coordinate or rather vector position of the centres relative to each other and calculates their common trochoidal path in 3d or rather in generalised coordinate representation.

It is brilliant!

I aggregate to measure.

Thus all my aggregation, manipulation, summation and calculation is to measure.

My algorithms, mathesis , ruminations and procedural rituals are so i can measure.

All my tools, preparations and conventions, directed and agreed practices, points of view and perspectives are simply so i can measure.

What and how i measure then is not solely up to me. and carries with it a ritual a convention a history and a mathesis. It has adherents, proponents antagonists and pugilists, as well as heroes and sages, lords and serfs, in short it is a thoroughly human enterprise, all for te goal of measuring, and through that process manipulating what is going on around and among us.

It is to develop a proprioceptive basis to our understanding of our existential experience.It is to measure the mystical and mysterious and make it mundane.

But in our measuring we make one condition: our unit whatever it is must be fixed, not slippery, still not dynamic, stable not changing. And that condition, understandable as it is , fails us and dooms us to failure and approximation; or rather, more cheerfully, provies us a launching pad to iteration, and an ever changing reality of lights.

When we aggregat we aggregate visually, and that gives us spaciometric magnitude but only in part. The other spaciometric magnitudes are bi-aural aggregations, bi-olfactory aggregations , and if we would let them grow a bi-whisker aggregation.

The close circumspect circumflex-kinesthetic aggregation  contributes to a grounding of all the other aggregates of intensity- magnitude. So magnitude that we measure has form and/or intensity or unending  unbounded space, which we can explore and investigate forever!

b              a*(a-b)/a
2b            a*((a-b)/a)^2
3b            a*((a-b)/a)^3
4b            a*((a-b)/a)^4
5b            a*((a-b)/a)^5    
 ....................
nb            a*((a-b)/a)^n

arithmoi          logos

The arithmoi is a geometrical form. In this case it is a rectangular cuboid, very thin, but increasing in length. It serves to record a regular plethoration of a unit 1b

The logos is the proportion written as a ratio.

Napier simply observes that a regular plethoration is in step with an iterative proportional decrease. The logos therefore had an associated arithmoi, it was the logos's arithmoi which melded to logarithm.

He had to draw attention to the arithmoi, because he noted that adding the arithmoi mutiplied the logos.

The idea of function is a later notion, but in everyway a table is the archetypical function definition, and Napier was developing a table. becaus of this he was able to develop a table look up notation that would define a function relationship in later eras.


Look up a certain value in the table and trace its logarithm:
 we write log(a*((a-b)/a)^n) gives n, and eventually over time it becomes =n




In this case a-b/a= sin(π/2-) being a very small radian value.





is used to work out a table of values for each n.

b              a*(a+b)/a
2b            a*((a+b)/a)^2
3b            a*((a+b)/a)^3
4b            a*((a+b)/a)^4
5b            a*((a+b)/a)^5    
 ....................
nb            a*((a+b)/a)^n
 Allows us to derive logarithms for any base =(a+b)/a .Thus for base 10 a=1 b=9





is used to work out a table of values for each n.





is used to work out a table of values for each n.




is used to work out a table of values for each n.


would imply a-b/a=+&-1=i
and therefore b=a(1-i)

For  an arithmoi +&-1 means that the form is a unit square with one side +1 and an orthogonal side-1 this gives a unit area of -1 which is the yoked unit to +1.

Therefore using b= +&-1=i

gives (a-b)/a=(a-(+&-1))/a = a(1-(+&-1)/a)

If i set a=1

1-(+&-1) becomes the base

The basic algebraic rule separates different kinds of aggregated arithmoi



also for
(a+b)/a=(a+(+&-1))/a = a(1+(+&-1)/a)

If i set a=1

1+(+&-1) becomes the base

The basic algebraic rule separates different kinds of aggregated arithmoi

Arithmoi allow me to understand units and scalars.

The greek arithmoi are spaciometric forms. The forms are magnitudes and to measure, a unit arithmoi is freely chosen and used to plethorate a form. The basic unit is a cuboid, and in fact the special cuboid, the cube has a foundational place in the proportioning of things, and the development of scalars.

From the cuboid are abstracted (or dimensioned): Area as surface area, and broken into directed face "vectors"; and length as perimeter and again broken into directed length "vectors".

A particular algorithm called in general rooting gives the unit length vectors of a cube form a common corner. In this case they happen to be orthogonal, but in general they are at the angles of a generalised coordinate system. The best way to visusalise roots higher than 3 are as roots of unity of a unit sphere.

Thus the solutions to the general polynomial theorem will  lie on the surface of a unit sphere, and these 'imaginary" magnitudes are no longer numbers but vectors in the unit sphere vector space. Some of them may form the corners of 3 dimensional polyhedra/polytopes.

Arithmoi.

For me the concept of form as magnitude has freed my mind. The concept of unit arithmoi has made sense of measurement and aggregation. the concept of dynamic magnitudes, dynamic arithmoi has enabled me to respond to a changing reality as a moton field , a fields of attributed and attributable arithmoi in relative motion.

Form is dynamic, and it is magnitude and it is in relative motion in a motion field of forms. And the motion fields of forms is boundless, in that i cannot see the boundary of it , and i am part of it and influenced by it.

And i can at last acknowledge a boundless structure as a fractal structure with self similarity at all scales, and therefore boundedness is not a problem, because there are always greater and lesser boundaries which i can attribute without tangible influence on what is going on around me in my appreciable and apprehensible environment.

So my Logos Response provides me with computed spaciometries in response to a surrounding signal source called notFS, from which i compute the set FS and my basic response is a "logos" field of dynamic arithmoi in relative motion or equilibrium.

Arithmoi are attributed forms: forms that my sensory meshes have computed boundaries and surfaces and dimensions and directions and rotations and reflections for. And sadly i had been taught to ignore the computational attribution of my meshes, and encourage to think of these arithmoi as properties of reality, not attributes of an incredible computational network.

So bit by bit i have been taught to make measurement harder and harder to understand, follow or achieve. I have been blinded to the intuition that i have computed all measurements in my sensory mesh already, and displayed the computation in all sensory systems already, and experienced he answer to any any measurement solution already.


Arithmoi are the beginning of greek reductionism of magnitude, Logos the beginning of greek appreciation of proportionality, atom the beginning of greek ideas on indivisible dynamic unity, and the foundations of their science, their mathematikos, their manthema and manipulation of the forms amongst which they moved.

 The greeks had epistemology and Sophia, but from Thales they learned empiricism, observation of how things relate, and they learned it from some of the oldest and pragmatic cultures in the world at their time. And they had more to learn: from India, from Egypt, from Persia from China, but they went the way of all ascendant cultures, but their legacy remained to be mangled by the victors.

Without greek influence the west would never have recovered and gone on to surpass in technological pragmatism much older and wiser cultures, whose practical sciences far outstripped western sciences. But by pooling these cultures wisdom the west eventually regained its ethos and went on through its forms of government and patronage to establish centres of intellectual competency directly related to the scholasticism of the near and far east.

Jews and Arabs principally played the major role of transfusing these competencies into the west.

So our western culture owes much to ancient cultural influences, but the difference that made the difference was greek influence transmitted also through jewish and arabic influences.

Arithmoi are dynamic spaciometric forms of magnitude  with attributions of  surface boundaries,and dimensions of volume, surface area and edge lengths. And dynamic attributes of form orientation and translation and rotation and reflection, translational and rotational shear, which dynamic attributions effect surface orientation, edge orientation, surface and edge relative rotations and translations and shearing, relative density and compaction and relative equilbria.

Arithmoi thus are dynamic and relative and reveal on study how aggregation is structured in measurement, and how units plethorate by ratio and scale ,and magnification reveals more and more about the aggregate structure of what is around us.

It has to be said that most of what we know today was known in the earlier cultures, but its application to technology required the western industrial revolution.

One man stands out  In my mind in all the World. Napier.

What he did has lead directly to so many modern mathematical , astronomical, and scientific tools and conventions and at least two major inspirations in thought which are fundamental to the way we view the reality of our day. Both Roger Cotes and Sir William Rowan Hamilton claim inspiration from Napier's Logarithms.

Logos and Arithmoi are absolutely fundamental to Vectors, versors, and Quaternions, which ling Greek and Indian trigonometry to the very quantum chromo dynamic description of Quantum Dynamics , to Einsteinian geodesic descriptions of space time via spherical trigonometry.

And these things do not have to be hard won, we can arrive at them playfully, because we have already calculated them in our sensory meshes!

Shunaya.

Somehow beautiful shunaya came to me as a woman, a rich indian princess,serious in purpose and bountiful. Her face is adorned with jewels and her hair braided into a tiara. Her gait is determined, and none who touch her remain, consumed into her like the finest of dusts, the most essential of oils and the most aromatic of perfumes. Behind her a wispy trail of stars in the blackness of the night sky slowly fades away into the morning light.

Look, but do not touch. Watch her go by in awe. Say her sweet name and taste it on you lips, for you will never possess her, and she shall consume you.

Shunaya

The finest of tilths, the essence of quintessential, the perfume of aroma and the merest hint of flavour, the thrill of the slightest touch, the mildest tingle of sensation , the whisker of a movement, he raising of the hairs on your skin.

Their are situations all around me which my sensory mesh cannot compute a boundary surface to, and yet my other sensors are computing their results to. The visual mesh computation produces no visible result to bound the magnitude in a form, so the other senses give their results as dissociated sensations, and yet not dissociated.

I have to remark that magnitude is predominantly a visual concept, and though i have referred to the other sesnors as also measring it i have pointed out that the measure intensity of magnitude or rather magnitude of intensity.

For a visual based person it is perhaps hard to accept the fundamental diference this makes to their apprehension of their reality. We all pay lip service to the blind person relying on their other senses. This is not an understanding, it is a concession , a way of saying you "see" through your other senses.

Think for a minute if you are visually based: can you smell through our eyes? Can you hear through your Eyes?

We watch a lot of videos, but we generally do not have a deep synaesthesia which enables us to smell a scene or hear a soundscape. We attempt to translate across through words and associated music, but this is an indirect fabricated experience not a direct sensory experience from or own unique sensory mesh.

Well here is a news flash. Some of us are not visually based! Some of us experience the world through the other sensory mesh taking the lead: some of us see the dead and smell the living and touch the face of God!

Generally then when a "substance" is too fine to compute a surface my sensory mesh can still produce computational attributes which are dynamic magnitudes of intensity with direction orientation and rotation and unit intensites just as before.

When i see a cloud, it is  a computed surface i attribute to it, but if i enter a cloud i may never know exactly where that surface is, but i can know the level of intensity associated with the other senses as i engage with the cloud. Who has not walked in a fog and seen it thin out around you but still look as thick in the distance, except a blind person? Who has not been in pitch blackness where you cannot see your hand in front of your face, but the intensity of the other sensory magnitudes lets you know how close it is?
Brahmagupta may not have originated shunaya or "negative" numbers, but he certainly put them both on the map for all eternity.

Shunaya is in its "majority" not computable or attributable to a surfaced form or set of closed surfaced forms of magnitude ; that is arithmoi.

Shunaya is however in its majority computable and attributable to intensity magnitudes, dynamic magnitudes for which i have not yet found a greek name, but prefer an indian name: maybe "shunayasutra" after the vedic verse that give advice on calculation.

Shunayasutra then are dynamic magnitudes of intensity with translation , rotation and reflection attributes. and yoked roots of unity. Their dynamic intensity reflect static and dynamic stabilities or Equilibria. And while it may appear that we have no trigonometry for them we in fact do through a transformation called homology. They are an interesting set of magnitudes to study and we in fact study them as fluid, electro magneto thermo gravito dynamics.

(http://nocache-nocookies.digitalgott.com/gallery/5/410_14_01_11_9_31_49_3.png)

There should be a logos-shunaysutra giving magnitudes of intensity a logarithmic assignment to aid calculation.

There are a few new view points: for example rotation is a magnitude that is attributable to direct apprehension by counting and usually shows no plethoration beyond the whole form and the sensation o spinning.  a rotating intensity would be very similar, and marked by the stability of the intensity.

Units of intensity would have to counteract one change present but not noticed in visual units, the tendency to dissipate through diffusion leading to a decrease in intensity. Plethoration would have to be distinguished from diffusion by measurement of intensity levels. Very likely plethoration is accompanied by diffusion in both intensity and visual magnitude units, which makes sense of a probabilistic description of unit position during motion.

As unlikely as it sounds we have calculated this already in our sensory mesh computations, but for each of us it is a unique measurement we come up with, and this helps define our uniqueness as individuals.

So where do sequences and series come from? Do they exist or are they tools of measurement devised in a fractal scaling tool by human animates?

For the Greeks the basic arithmos i think may have been a plane figure. For me , however the basis arithmos is a solid.

Euler decomposed a solid into faces edges and vertices.

My first observation is that this is a basic polynomial aggregation.

So if i add the nomial  solid to the list i can write S for solid, F for faces, E for edges and V for vertices  .

Thus a cube aggregation would be


If i adopt the convention that V stands for Origins, E for vectors from origins and F for vector sums from those origins in 2d and s for vector sums in 3d i have a vector polynomial, or a polynomial which could be similar to a polynomial in 3 variables.

This gives say



where 0 is of course shunaya and so contains yoked roots of unity .

When we set a polynomial = to (0) shunaya we are in fact posing the question, what are the roots of unity of this solid? In a plainer explanation: we are asking which edges are connected to each other through a vertex/ corner?

The answer for a cube will depend on us identifying the edges, but what we do instead is identify the unit measure, so the answer is strictly 1and1and1 or 1&1&1.
This is  a yoked triple root from shunaya and therefore has 8 variants if we introduce Brahamaguptas yoked unities ±1 . 8 comes from the 2 choices(+ or -) and the 3 yoke giving 2³.

Why do i have a choice  of say red or black, fortune or misfortune,+ or - ? As indicated this arises from the application of these vectors to everyday life. We chose to give this initial yoke this choice significance. Thus if i choose to signify it as a rotation through π radians, i can as long as there is a mod (2) aggregation arithmetic underlying.

So i could use sin(π/2),sin(3π/2) or cos(π),cos(2π) because these are mod(4) relationships that can be made mod(2) by using P radians, but not sin(π/2),cos(3π/2) because the mod(2) relationship is out of phase. However sin(π/2),cos(π) can be used.

The relevance of these significations is that whole branches of scientific significances rely on them! They even include what some cal pseudo or irrational sciences, but which clearly come from the same root ideas!

For me, it is of significance that "imaginary numbers" have defeated their derisive jibe and shown themselves to involve and engage the imagination of human animates in ways that have liberated them from the tyranny of "correctness" , conformity and convention, and the encrustments of coercive power elites who seek to enforce what is unenforceable: their unified view of perfection. Unfortunately Pythagoras falls into this category.

If we are to learn anything from the "imaginary numbers" it is to let human animates imagination roam free.

So clock arithmetics, that is modulo arithmetics describe the ratios of rotational magnitudes, amongst other things, and reveal phase differences which we have come to denote by roots of unity. These phase differences are ratios linked through the trig functions to logarithms, as are all ratios/ scalars.  

Modulo is therefore linked to logarithms by this means
And the meaning of roots of unity are found in the logarithms of trig functions. But the method of Napier links series: arithmetic to geometric; and so he encompasses the foundations of Calculus and one can easily derive calculus formulae by his methods, including Taylor, Mclaurin and Cauchy series expansions.

(http://nocache-nocookies.digitalgott.com/gallery/5/410_14_01_11_9_31_48_0.png)

Cotes (http://www.mathpages.com/home/kmath192/kmath192.htm) easily derives Euler's formula from studying Napier's Logarithms, but Napier did not use base e as is assumed by many.

Cotes may very well have derived e as the base of his logarithms decades before Euler, but what is of interest to me is that Cotes derived his formula in the context of astronomy and Newton's work on gravity. He died before he had time to explore its significance for astronomy and Newton's Laws of gravity.

(http://nocache-nocookies.digitalgott.com/gallery/5/410_14_01_11_9_31_48_1.png)

I suspect he would have put it to Newton that his Laws were a first order approximation of a much more general Law of gravity.

(http://nocache-nocookies.digitalgott.com/gallery/5/410_14_01_11_9_31_48_2.png)