Fractal Foundations of mathematics: Axioms notions and the set FS as a model

[jehovajah]

Thanks Herrmann, great illustration of the concept.

I have no hesitation in saying that Bombelli preceded Descartes in using an orthogonal coordinate system.

"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  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.

[jehovajah]

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!

[jehovajah]

By analysis found out basic orbits are trochoids. Have not worked out an interesting 3d formula yet but this is a start.

[jehovajah]

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!

[jehovajah]

Finally   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.

[jehovajah]

Of course i am not going to stop posting interesting tid bits 

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.

[jehovajah]

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

[jehovajah]

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.

[jehovajah]

Among the scalars of unity the primes 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 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 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.

[jehovajah]

I have in fact turned up a new hero in the story, a certain Roger Cotes, 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.

[jehovajah]

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.

[jehovajah]

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 which he took to represent a vector with a scalar time component.

Thus the 3d vector is given by  <Quoted Image Removed>

"The quantity
<Quoted Image Removed> (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
<Quoted Image Removed>, so <Quoted Image Removed>  
Then
<Quoted Image Removed>"

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.

Hamilton immesiately refers to these as a system of equations, 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 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.

[jehovajah]

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! 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..."

[jehovajah]

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!

[jehovajah]

Spherical trigonometry provides a link to quaternions.



As you may read, the application of quaternions to special relativity and topology is fundamental. These topics 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 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 therefore will give insight into the "workings" of volume vectors and provides insight into their interactions. 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 i could not think of if i tried!

The Lorentz boost 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