Der Ort der Hamilton schen Quaternionen in der Ausdehnungslehre

[jehovajah]

Frabjous joy! Kaloo Kalay!

Scanning through the 1862 version to find the definition of the closing in product has been interesting. There is little doubt that the outer product contains within it the products or multiplications of the roots of unity. This presentation of it may perhaps be unfamiliar to many, but no wonder, few today study the intriguing and mesmerising theorems of Cotes and DeMoivre for the roots of unity. They were at one time of great mathematical interest and tHe basis of many new number designates. I can think of the elliptical numbers as an example, besides the general complex numbers.

The research and work of Robert is evident in this treatment, which shows an advanced knowledge of higher mathematics, something Hermann was hard pressed to keep abreast of during the writing of his first version. Indeed Hermann admits that he did not even know of the Gaussian form of complex number notation in 1844, 13 years after Gauss had published on the subject!

However the reason for my joy is that the definition of the closing in product is as I have adduced from the translation. To say that it is ridiculously simple is to decry the complex use that Grassman puts it to, and the generalisation he exposes by it! However, it is ridiculously simple!

As I suspected the terminology " inner" is misleading which is why I banish it. The closing in product does one simple thing: it introduces distributive multiplication of bracketed forms into his algebras.

Given any arbitrary line segments emanating from a common point and spreading apart, there spreading apart product is the 2 line segments projected by each other so as to form a parallelogram.with the intersection of the projected parallel lines identifying a fourth point diagonally opposite to the point of emanation. The product is written simply by the labels for the bounding line segments put together in order of the principals of orientation and direction.

Thus if the principal orientation is horizontal and the principal direction is left to right, and the principal rotation is anticlockwise we must write for the line segments a and b  ab, providing a is horizontal and is drawn left to right . If we now write ba we imply b is horizontal and drawn left to right and a now has a relative orientation .

In this light it is clear that ba would not attract a negative sign without some other rule bring in place. This rule is cyclic interchange.. Cyclic interchange requires the line segments to circulate cyclically around the form. In a triangle cyclic interchange does not attract a sign change. However in a parallelogram cyclic interchange attracts a sign change that is reminiscent of so calle complex number multiplication!

Having established these rules for the spreading out product. Hermann established some simple rules for the sum of line segments.( historically this was his first set of rules). These rules are best expressed in poit form

AB + BC = AC .
In the case of the parallelogram product this must become AB = a AD = b AD projected parallel by AB = BC =b , so AB  + BC = AC  can be written as a + b = c, where AC is now labeled as c givin it relative orientation, and principal direction.

Well this has several notational difficulties, especially the sloppy use of= sign, and the visually clear but algebraically obscure identification of AD as abelled the same as BC !these difficulties cannot be obscured or finessed away .

The identification of the "equality" of parallel line segments has to be properly understood not in terms of " equality" but in terms of duality or congruence.

Duality is a fundamental Pythagorean notion( isos) and it is usually translated as equal , but in fact it does the notion a disservice to tie it to equality. In fact the phrase "liberty fraternity and equality" reveals how slippery the term equality is in everyday use!

As a consequence of this slipperiness we can determine where definitions have bern put in place to tempt to fix a meaning. By the same token we can appreciate where those definitions fail to capture our experience of " sameness" or congruence or identicallity!

In the work that Apollonius does on geometry, and the projective geometry of DesArgues and others duality becomes a defineable property. The best explanation of this concept I think is found in thr Universal hyperbolic geometry of Norman Wildberger.

So now we come to the closing in product! Up until this point the distribution of multiplication over addition has not appeared, but as soon as a perpendicular is dropped from one line segment to another the closing in product appears! It is called the closing in product because the line segments must be closing toward each other( or towards extensions of each other) for the perpendicular to be drawn. Once the perpendicular is drawn it segments the other line segment into 2 . Both these line segments can be producted with the "dropping" line segment translated to this intersection point. This gives the original product of the bounding line segments as the sum of 2 other products of the appropriate line segments. This is the Resilt of dropping a perpendicular and this result itself is the closing in product.

It soon becomes clear that there are many other results that sum to the " original" or spreading ot product, and in fact all of these results can be defined as a closing in product. The closing in product defines the distribution of multiplication over addition in group theoretic terms. The unfortunate word inner conducts the mind elsewhere and obscures this simple realisation.

The realisation of this fact frees the constraints of " perpendicularity" . Now the projection does not have to be vertically into the other, because the distribution rule does not require it. However, the useful rule that when the closing in product disappears the line segments are orthogonal also has to be modified. This rule does not in fact hold as popularly stated. It requires several conditions to be "true". The closing in product is intertwined with the spreading out product and both must be consulted to determine relative orientations. In particular, some have loosened the strict principal rules Grassmann and his peers worked by, making it relative to the observer but more confusing to the student.

Grundsetzen or ground rules are absolutely essential to any cooperative human effort, and the fundamental ground rules since Wallis have been the orthogonal ordinate coordinate system! However prior to that DesCartes introduced the fixed generalised coordinate system, often ascribed to LaGrange, but in fact originated by DesCartes. The essential rule was that all should agree which terms were fixed before the discussion or demonstration began. These principal terms and rules defined the conclusions. Time must be spent communicating with colleagues to ensure foundational consensus. This is why conventions are important.

Because many experience confusion because the conventions are misunderstood or not known; or they do not realise that the conventions are agreed rules which one is at liberty to modify, many become disenchanted with the sophistry of so called Mathematicians, and thus with mathematics. However, the fact is that much of what is taken for granted as truth is of the same nature as mathematical conventions! Thus a vigorous mathematician ought to be able to explore the system and rules that govern his determinations, and thus ought to be an ardent philosopher!

Admittedly it may at first be uncomfortable to exist in a thought world of no certainty, only definitional propositions, but if one pursues this experience one gains great freedom. On the other hand one has the constant practice of forming ties or bonds of common terminology with ones colleagues. If this aspect is not done then one may as well be as mad as the March Hare for all the good this formulating will do you!

The closing in product therefore in hermanns day and mind tied the general spreading out product to the conventional trigonometry of his day. By working in the spreading out product mode he could simplify many manipulations of spatial forms, only evaluating at the last few lines by reducing the formulations to the closing in product.
How he does this is naturally the subject of the Ausdehnungslehre!

How this process works for the description of Quaternions is the purpose of this paper. The first thing to note is that a mixed or median product process is required! Because we are told that multiplication is based on tables we cannot grasp what a mixed or median product is!

The notion of multiplication must first be removed, replaced by the notions of a geometric product process, founded upon factorisation as a division or quotienting process, and then we can begin to understand what we are doing in manipulating space! Numbers I am afraid have obscured this fundamental role of the Arithmoi!

[jehovajah]

I have noted how Grõße and Stüfe have replace Strecke as fundamental primitives in Hermanns method of Analysis and synthesis by the 1862 redaction. Größe as a fundamental experience of extension is self evidently a fundamental primitive. One must bound Größe to get quantity.

However it is Stüfe that attracts my attention. This of course means to me a rising step. So the concept is of advancing to different levels . This is appropriate to a concept of rank, so that at each step or stage we rise to a different rank. Thus we can put all quantities into ranks. For example the rank of a point, then a line, then a plane, then a solid and then ranks of crystals or more complex solids.

The system of monads or Einheiten( rather Arithmoi) used to measure or quantify and distinguish ranks naturally must be magnitudes of independent orientations in space.. This is why the line segment was such a fundamental quantity initially.

However the notion of experience of extension and experience of rank difference must philosophically be prior to specific instances of these concepts. In other words we must have a prior sense of rank to be able to identify rank and rank differences. This notion is as simple as experiencing a rising step.

While a magnitude( Größe) is experience of extension, we also have ability to experience and decree limitations. Thus this ability underpins the notion of a staged or limited rise, hence "step." or rank.

With these two philosophical notions of a priori cognisance in place we can define sums or combinations of products as products of quantities. Also we can understand these combinations as sequences of products which form or describe ranks of quantities.

Each product itself may simply be a term for a form. A sequence of forms may provide ordinal information only. It is the combination of a sequence of terms or forms , where the forms are clearly at different ranks or independent orientations or both that fully describes a rank!

Clearly when a rank goes above a solid into a crystal form, it is possible to reduce a rank to a solid one. Thus a crystal , being a 3 dimensional object may be fully described by a system of monads involving 3 independent steps. However it may be initially easier to describe it in a higher rank form and then to see if it can be reduced to the solid rank, and what precisely that entails!

In this case it is fair to describe the rank as defining the notion of dimensions in space. Such dimensions are simply convenient orientation.

 However once again popular ideas of dimensions have been so misconstrued that we fail to recognise the n imendional nature of the space in which we live!

There is one other important aspect of Hermanns method. Since it is quite general the definitions can apply to analogous quantities which at first glance might seem qualitative rather than quantitative. However any experience with an extensive quality can be quantised!  Newton discusses how this should be done, using the invariant product forms of Spaciometry to do so.

The use of product forms for measures underlies Normans concept of spread and quadrance. While at first thought it seems a funny way to be measuring, in fact it highlights how screwed up numbers have made our perception of what we are actually doing in space!

An interesting comparison in the word step shows how Hamilton used the notion to distinguish a stride length in a certain direction, while Hermann used it to determine rank!

[Roquen]

From a quick skim (I don't have all of these paper anymore) I'm not finding the version of Hamilition's rotation.  The Altmann paper seems to refer to it however.  I should note that I read these around 20 years ago, so it's insured that my brain has reinvented some parts. I want to remember he was breaking the equation into part (somewhat like how a euler-angle formulation works) so he could use P'=RP like forms, which only work iff R and P are orthogonal.  However I did run across a paper "ON QUATERNIONS AND THE ROTATION
OF A SOLID BODY" from 1850 where he uses Cayley's form.

I've often wondered how Olinde Rodrigues's paper became ignored.  The reason here is that Cayley not only cites it in his paper...his demonstration that the quaternion equation is a general rotation is by showing that it's equivalent to Rodrigues.

Another player in this time period is Louis Poinsot.

[jehovajah]

Roquen, the link you posted briefly refers to Hamiltons " broken" first formulation, as well as the general formulation.  It is clear enough to me that quadrant rotation is broken by his formulae because he uses the the principal function labels i,j,k instead of a pair and a conjugate k for example.

However, many more constraints then become necessary to establish a division algebra, and the whole unbelievable process ( at the time) becomes unwieldy and artificial! Part of the mystique of Quaternions was their extension of ordinary higher arithmetics without disconcerting fuss and the attraction and comfort of Pythagoras like "squares sum". In addition Hamilton invented a labelling that indexed certain standard measures and made handling the expanded brackets seem easier.

However at this juncture it is Cayley who comes up with a tabulated format( matrix) that makes the expanding brackets process more accessible than the multiple summation products!

I am pretty sure that the sum notation was a welcome innovation when first introduced, but there are many rules to its use that simply get bypassed in today's education.  Moreover , when producing sums the notation though concise , and revealing at the term level gets in the way of intuitive manipulation. Cayley's table products restore an intuitive feel to this aspect of expanding brackets.

(* in the 1862 version the sum notation is developed extensively from the outset, but section or statement 42 and Statement 100 bear directly on the innere product rules. While a Cayley table makes this so much simpler to explain there are a lot of associative manipulations that are ignored. These manipulations and mental facilities make Grassmann seem very subtle to the modern reader. However it is simply that we are unfamiliar with many standard associative manipulations in the rules of distributive multiplication over summations. These are what Grassmann referred to as the multiplicative product to the addition!)

It is worth a study.


When I was introduced to Matrices I was not given the table concept. In fact I saw the relationship between a product table and a matrix only fleetingly, because I was introduced to coefficient " matrices " first.

When later the determinant, the transpose and the sign rules of a determinant were introduced, I was already lost! The ad hoc appearance and relationship to ordered pairs of coordinates, n-tuples, vectors and Tensors was not a winning presentation for me.

Later the ot product, cross product, vector space combinations in set notation( direct sum)  seemed verbiage for verbiage sake. So now I have finally grasped the dot product, due to Normans consistent presentation of it, I can see clearly how it arose and how it relates to expanding brackets, product of sums manipulations and the Closing in product, and eventually how it is underpinned by a non geometrical , non Arithmoi " multiplication" operation.

Everything that early 19th century mathematicians before Dedekind developed were based on the Arithmoi of Ruclids Stoikeia and the Logos Anlogos calculus of homogenous magnitudes. Books 2,5,7 as fundamental grounds. Suddenly this was all thrown into question as new nonEuclidean geometries were touted. Even Gauss was reeling under the massive failure of mathematical geometers to solve the parallel postulate problem, conveyed from east or Islamic scholars to the west by Legendre.

The point is, and do not be deceived, this was a centuries long mistake of the design and structure and purpose of the Stoikeia course! There is no parallel postulate! The demands of the course required students to be technically and mechanically competent. They needed to be able to draw straight lines of any length( and that is not easy!) circles of any radius or diameter , and to enure that if a line was not at the same angle to a transverse line as another,,both in the same flat plane, that those straight lines could be accurately drawn through their intersection no matter how far away that might be!

Finally it was required that the accept without question that all right angles are the same?( Thales theorem for angle corner sub tended by a diameter at the perimeter.

Parallel lines are a defined concept found in the first definitions. The fact that other lines and curves can be joined in a kind of curved triangle was well known and well explored in Conics and spherical geometry, and spherical trigonometry! Lobachewsky, Bolyai and Gauss "disvovered" no new geometry or Spaciometry, no non Euclidean geometry, because what they found was already known in spherical geometry.

It is just one of many embarrassing periods glossed over in mathematical history.

So Gauss and Riemann and others lost confidence in Legendres interpretation of the supposed foundations of geometry. The academicians had lost their way. In that regard Grassmann restored confidence in fundamental Euclidean philosophy by tarting at the very elementary stages and gradually constructing the whole of higher mathematics on a few consistent principles nd a wonderfully useful notational technology.

Becausenhebdid not shirk from hard work he went step by step through every aspect of geometrical development of products, including the use of the summation notation.

The dot product and the closing in product are Clifford's identification of essential stps in the process of expanding brackets . These stps are made graphically clearer by using Cayley product tables.

Using Cayley product tables a variety of products as us of products can be defined and identified. It is precisely these products that Grasmann uses in his manipulations of product sums , summation products and the establishment of propositions in his system.

The dot product defines how 2 rows or lists or columns are combined to form a coefficient-basis  sum which is at heart any general product sum. This process is used again and again, but each,reapplication requires the product sum to be changed into row or a list or a column first.

To be strict this has to be designated by another process symbol, or a complex symbol. Grassmannmthought it necessar to introduce the divided Bracket symbol to represent the closing in product. Within that Clifford introduced the dot product as the initial stage of a compiles product sum construction best described by a Cayley product table with summation rules.

In Quaternions the different products have a well defined Cayley product table position , but one has to recognisenthebdot product atnabdifferent rank orbstepmfirst.
Thebcoeffcientsband the monad system arrangedbinmabcayleyntable produce more products than the required product , which is formed by products along the leading diagonal summed. These other products can be used for other purposes, but they simply are ignored.

Then using this dot product turned into a row and producing it with another arbitrary dot product turned into a column, another Cayley product matrix can be formed, from which a dot product can be constructed. Clearly this second dot product is at a different level or step or stage in the process!

At this stage the other productsvare not ignored. They now are used to form the vector combination sum, the cross product vector combination sum, so that a quaternion product can be defined as a combination of distinguished products, each separated by a label marker, or a list format or a component format( row or column). The terminologymvectornis very loose here . Some may call these formats vectors or tensors. Few will draw your attention to the position in the quaternion Cayley table! These insights are buried away in Matrix theory!

While studying Grassmanns notation it is well to remember the importance of Cayley tables. They are fundamental general layout structures for developing a number of processes from reference frame synthesis to matrix and determinant algebras, to computer memory structures . But also remember Cayley tables are way of representing processes of complex construction and production going on in the computing minds of Hamilton and Grassmann,,Gauss, Cauchy and even Japanese and Chinese savants. They are as fundamental as layout on a sheet of paper!

[jehovajah]

Of course, the method of construction Grassmann was using is more general than the Quaternions. Thus the terminology of the Quaternions does obscure these more general processes. Whle we may define a dot product for any Cayley table, the vector combination and the cross product combination only appear in the quaternion level. For this reason the more fundamental Wedge product is preferred by Clifford Algebras.mthe wedge product is a recursive or iterative product . It turns out to be connected, via the closing in product with the evaluation of the spreading out product, and the adeterminant of a matrix where defined.

Norman insists on calling it the cross product for obvious reasons, but it is not to be confused with the vector cross product

While table formats are used to define the wedge product it must be notednthatnthesenare closing in product tables not spreading out product tables. The spreading out product is associative not distributive. Any distributive table associated to the spreading out product will,be a closing in product.

Any combination of spreading out products will ultimately be a closing in product of some higher ranking or higher stage spreading out product. The aspect of reducing rank or stage to a step which is fundamentally evalateable forms a big part of sprading out product manipulations.

[jehovajah]

The closing in product in 1844 was conceived in a dynmic geometrical construction. This involved dropping a perpendicular, or projecting  vertically one adjacent line segment into another. The right triangle so formed symbolises the introduction of a reference frame.

A monad system consists in iindepent collection or set or sequence of " independent" Metrons. Metrons are experiences of extensive space which are bounded, as an inherent part of the experience. Thus each Metron is a bounded unity or unit.. By these we count , and thus quantify , those extensive experiences.

A Metron as a bound experience may have a visual component we call it form, but not necessarily. Each Metron/ monad is " independent" if it cannot be counted by ( using) any other Metron. Thus all Metron/ monads in a monad system are unique or " protos" . This means the first of its kind or type, but it is also stated as " prime" or relatively prime.

We can thus construct a monad system from prime Arithmoi , "protos Arithmos" if we wish. If we try to use the number concept here we run into all kinds of confusion, but in fact we can use Modulo arithmetics, that is groups of " numbers" with a modulo rule inherent to model such monadic systems. They are also called finite groups placed in direct combination or cross combination with each other.

In the Stoikeia these monad /Metrons were any of the standard forms/ ideas drawn and studied in the plane or in pace. The modern numerical concept or Area is a complex reduction of all these flat forms into one standard form ( square) or similarly for volume the standard form is a cube. Thus in a direct geometrical analogy the concept of a closing in product is the concept of reducing all or any form to an equivalent standard monad/ Metron mosaic, or mesh .

We can usefully analogise the spreading out product as any of the many different rocky crystal shards we see in a gravel pit for example. Here is where it gets interesting. Using these crystalline shards many mosaic forms and scenes were embedded into the floors of houses and temples. Such complex mosaics helped to establish the notion of the Arithmoi as fundamental natural nd supernatural constructs. Thus the Pythagoreans studied these patterns intensively.

It is clear that certain rocks are not jagged crystalline forms but indeed smooth ounces forms or pebbles. These properties make very little difference to the Philoophy ofnthebarithmoi, but the modern concept of area runs into some difficulty when making measures of space using these different metrons. Such a problem was and is cosmetic, but it took the concept terms of Benoit Mandelbrot to shake a centuries old mistaken geometry, and re admit the notion of scale free recursive or iterative applications of a irregular form. The word fractal wasmsupposed to describe how rough a geometry was. This was to use not precise regular Metron self similar at all scales, but rather Metrons that were "almost self similar" at every scale. This last self similarity was to be measured by a curious counting method to give a ratio or numeral which was to convey how roug the fractal geometry was!

This is a burdensome notion! But mathematicians are like that. If some scale of quantitive measure of a qualitative " feel" could be agreed then no one would doubt the mathematical nature of these " geometries".  Fortunately artists took over the development of these ideas and created a graphic icon that now bears his name and shows the dynamic properties of spatial Metrons . The convoluted fractal dimension measure , still pursued by mathematicians is pragmatically replaced by aesthetic assessments.

It is perhaps an interesting historical study to track exactly how artists advanced the development and acceptance of computers more than any other Academic group!

Nevertheless a monadic system consists of combinations of these various Metrons including spherical forms.Grassmann called these combinations Größe or magnitudes. These magnitudes encapsulated partial fom and also spatial dynamics of form transformation. but they are still pretty mysterious and experimental models of our experience of form and dynamics in space! Because we have a reliable symbolic format, a terminology that appears to be trustworthy many have moved from,spatial transformation and manipulation of these magnitudes to algebraic shenanigans!

However, the painstaking research and development of this terminology and attendant rules of application and applicability which we rely on were substantially achieved through the part time efforts of Hermann Grassmann, who constructed lineal Algebra from the intense study and redaction of the woks of the geniuses of his day with an inisght he felt was only given to him! This was substantially true at the time, although Möbius came close , and Hamilton closer still.

St Vainant apparently came up with the same insights, but this was judged to be an act of plagiarism while translating into French Hermanns key ideas!

The closing in product is defined as the product of any arbitrary Metrons or monads that form part of a monadic system. The important point that has been generalised from the early vertical projection insight is that the projection introduced a reference frame into the description of an arbitrary extensive magnitude. The closing in product defines how we are to manipulate the components of that reference frame .mthebconstraint isbthatnwe must get the same result using the spreading out product and the " underlying" closing in product.

This is the concept of invariance: Keine Abweichung!

This is virtually guaranteed by the definition. The same product rule applied to the monads works through to,apply to the combination magnitudes. At least formally the notation means the same at the level of a monad product as it does at the level of a magnitude product. Thus the closing in product is scale free in this fractal sense , and this is why our methods produce these almost self similarities.

Am I saying that we somehow create these fractal self similarities? Yes. Any system which relies on our mathematical models solely will return selfnsimilarnresults at all scales.

That is why mathematical physics etc without empirical observation can not give me a " true" absolute picture of reality. It merely gives me a model, one of many possibilities. Any one who studies Grassmanns notations will recognise how it influence Paul Dirac and his presentation of Quntum Mechnics, for example . Where the application of identical notations differs is in the empirical interpretation tied to each and every jot and tittle of the notation!

The closing in product, as Norman distinguishes, introduces metrication into any general geometric or rather spaciometric description using a standard format. The SI unitsmfor example form a monadic/ metro system for metrication of our geometrical descriptions of forms dynamics and pressures and powers in our physical world. As such, how we tie everything back to these units is our closing in product method or system of methods.

[jehovajah]

Returning to the 1844 version to seek the closing in product reveals a different focus and exposition entirely. The Abschattung product and the eingewandtes  product are the main counter points to the spreading out product. In addition the spreading out product is used to represent the 4 main operations for addition division multiplication and subtraction.

The Abschattung product and the eingewandtee product I am now going to explore briefly. Despite his general outlook and clear effort to meld together his specific and general ideas, the fields from which he drew inspiration and applications for his ideas are overpoweringly evident in the layout. Newtonian kinematics, harmonic analysis, imaginary products and projective geometry are all mashed up together in this first volume !

[jehovajah]

It turns out that the eingewandtes product , a major theoretical plank of the first volume was a dud by the 1877 revision of the 1862 version. The product which I am going to call the special pleadings product or the postulated product, requires a number of constraints to be true or fulfilled in order to determine it safely from the spreading out product and the null result.

This kind of detailed analysis relies on the concept of a system of equations or expressions by which certain relationships of quantities of magnitudes are established. The image or idea is that the spreading out product successfully manages or manipulates such a complex system of equations so that you can multiply or divide them as systems . The full set of operations addition and subtraction included apply to these systems.

While this is now dealt with as Matrix algebra, it is in fact first exposited in this complete way by The Grassmanns ad Hermann in particular. Cayley get the credit for sweetening the notational form so that all those summation signs and operations become more intuitively manageable as a product form. However this form exists as labels in Grassmanns work in the spreading out product, but it is so much clearer to represent the summands by matrix notation, or rather Cayley table notation.

However, the analogy is severely constrained! It is these constraints that lead to the exploration of and naming of different products. The postulated product is such a distinction. You must remember that a postulate is a special pleading, not a proposition as some still find it hard to understand.  Thus this product of all the products demands the most attention!

The product requires so many constraints to be in place that I am not surprised and thankful that it failed to be a fruitful idea! However at the time Hermann went to great lengths to plead for its existence, demonstrating how it should work in analogy to the spreading out product.

However in a footnote made in 1877 Hermann quickly alerts the reader to the fact that it was not a fruitful development in the method but that its ideas we're absorbed back ino the general body of the method. Although I have not looked it up, the regressive product suffers the same fate by 1877, as Hermann seems to identify them as the same kind of special pleading product.

What this means is that by 1862 the Abschattung product had become the main counterpoint to the Spreading out product, and it is there that we might trace how the closing in product achieved it form as expansion of brackets, which brackets represent the magnitudes written as the sums of monadic/ Metron systems.

As I pointed out earlier the dot product plays a crucial role and the closing in product as column row Cayley table products simplifies the Process considerably.

<a•A|b•B> is the forerunner to the Dirac Bra and Ket system of notations, and the tensir notation of contra and covariant tensors( n tuples rather than 3d vectors) and the Einstein summation conventions! All of them trace right back to this seminal work in 1844 in which the spreading out and closing in products were first identified as geometric products and then instantiated as algebraic or symbolic arithmetic forms. The chief foundation for the closing in product in 1862 was the Abschattung product, but in 1862 it took on the general Einheiten product format, which always from the outset in 1844 allowed it to adopt the most general form.

It is perhaps helpful to review Normans videos on polynumbers and on matrix multiplication. However one thing to note is that the [|] or <|> notation represents a summand of the Cayley product tables, whereas matrix multiplication represents a Cayley product table of dot products! The Cayley product table does not actually have a specific product symbol but it is a fundamental combinatorial product table and deserves one I think .
[ # ] where [a#b] is the Cayley product table for list a and  b in row column format. This could then be used to describe Hermanns combinatorial monadic system or Einheiten products.

These products as distributive rules are defined for any lists, magnitudes as monadic systems , rows columns etc, but the dot product is only defined for one to one lists at present.

[jehovajah]

This charming video is advanced conceptually. I place it here because the concepts ultimately reflect Hermanns most general ideas. At this level the discussion is algebraic and general. The algebra is then given possible interpretations in a consistent or coherent manner by empirical data, observable phenomena.

I might add that the concept" time" though emotionally locked in in our culture, is in fact a circular dynamic Metron . The monads we use are periodic motions, metronomic : like a pendulum, a metronome or a regular orbit.

Google "jehovajah time" for a deeper discussion.

Crucially time,Tyme etc are simply concepts of change in position. Thus displacement is the fundamental notion of both our concepts of velocity and time. Change in displacement is inversely proportional to change in period!  We cannot subjectively distinguish these experiences without this inverse experience.

When I am busy time slows down. But when I am not busy time speeds up. Inversely my perception of time speeds up and in the second case it slows down! It slows down when we watch it , speeds up when we watch things in complex motions!
<a href="https://www.youtube.com/v/oy47OQxUBvw&rel=1&fs=1&hd=1" target="_blank">https://www.youtube.com/v/oy47OQxUBvw&rel=1&fs=1&hd=1</a>

[jehovajah]

It always struck me as curious that I was asked to solve a set of equations by setting the equations to Shunya!

Shunya means everything so I guess you could say that means the equations can fit everything! The alternative is the equations fit nothing!

We really use the additive identity and the multiplicative identity to solve equations. Thus 0 Shunya and 1 monad the indian and the Greek notions of the alpha and the omega of everything. So when did -1 become part of the solution set? When negative directed number were introduced!

[jehovajah]

While this is not one of Normans most helpful presentations, and contains several confusions, nevertheless it illustrates how applying the Grassmann method requires careful consideration of constraints and interpretation.

The butterfly diagrams are only of interest in tht they show how geometric conservation of Arithmoi count underpins the motions of conservation for any extensive magnitude, especially those symbolised by line segments.

<a href="http://www.youtube.com/v/OAKZ7l3O92s&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/OAKZ7l3O92s&rel=1&fs=1&hd=1</a>

The clear and rigorous use of analogous thinking is the hall mark of Hermanns heuristic method . Because of this, symbols like line segments take on a powerful anchoring role, being, becoming and giving meaning to concepts and ideas that exist only in the expositors head.

[hermann]

Thanks for the Link to the Video: The Dark Side of Time!

Hermann

[jehovajah]

I hope it was thought provoking, Hermann.

I have had a long standing project to deconstruct the notion of time and replace it by something more appropriate. The link between motion and spatial experience is fundamental to my notion of order . The sequence in which my experiences of satial extension and cognisant focus is laid down in my memory of perception basically is my consciousness of " time" as a dance of processes. How to symbolically represent that is one thing I am gradually learning through hermanns work , redaction and philosophy in the Ausdehnungslehre.

Heuristically he builds it on the thoughts and meditations of some of the greatest geometrical, mechanical and philosophical thinkers of humanity.

Winces steady deconstruction of these concepts is interesting because he relies on symbolic processes to models aspects of our perception and constructs a notion of process which is essentially circular or fractal. These are his distinguishable elements from which he constructs 3 d time.

Normans example is interesting in this regard since spacetime graphs essentially record velocities as slopes, but mass appears as a line segment that does not move! By extrapolation in a 3 d system mass would appear as a volume that does not move, but whose volume projects sloping spaces in solace time . This extension or radiation of mass into space time is conceptually difficult as a construct, but it contains our notions of conservation of mass, energy and momentum, geometrically expressed.

How do we interpret that in our daily experience?

[jehovajah]

In the 1844 version the Abschattung product is discussed in the light of the notion of projection. But interestingly the larger discussion was about the grounds or geometry!

It is not until 1853 about 9 years after Möbius and Gauss had seen Hermanns text that Gauss instructs Riemann to make the Grounds of Geometry the subject of his Habilitation speech. The irony was not lost on the academics of the time, who later found the history of the 1844 text very compelling as a source document!

Be that as it may I am still in the process of reading this chapter to determine its contribution to the 1862 closing in product concept. In so doing I notice how several themes have been rearranged and given different emphasis between 1844 and 1862, as Robert and Hermann perfected the exposition.

[hermann]

jehovajah: I hope it was thought provoking

It was more then that, I have so many idears in my mind that I have difficulties to write them all down.
For example the problem of matter an antimatter in our universe. On the macro scale we see only matter.
Antimatter can only be produced in acceleraters like Cern or through radioactive decay.

An very old idear of mine is, that at the big bang the same amount of matter and antimatter is produced. The antimatter is travelling into the past and matter into the future.
This allows to produce two universes out of nothing. It is only a rough idear and it is beyond may capabilities in general relativity an mathematics to workout it out as basic for a scientific discussion.

From this point it is also possible to think of an oscilating process of matter moving into the future and antimatter into the past.
At the present we experience a stream of matter moving into the future. May be we have simultaniously a stream of antimatter moving into the past, forming the basic construction of spacetime as we experience it today.

I personaly dont like the following idears:

• The universe started as a singularity.
• Inflations of the universe.
• Dark matter
• Unsymetrie between matter and antimatter in the known univers.

Hermann
P.S This is only a rough sketch of some idears.