Der Ort der Hamilton schen Quaternionen in der Ausdehnungslehre

[jehovajah]

As I reluctantly come to the end of the induction, knowing that I have next to tackle the 3 product types and the development of the mixed product rules in this paper, I feel that I m returning to the chos that is modern mathematics.

The philosophical treatment of contemporary mathmatics, hermanns contemporaneity, and yet correspondingly our time has been eye opening.  The move from the philosophy to the mathematical process so called is one filled with trepidation! The Herculean work involved in setting all processes on a sound footing is daunting. And it is not as if I have not been on this journey for a long long time, but rather as if I have found what I was looking for but now have the task of mining the riches of the gold and precious metal seams I have uncovered.


At the last my childlike vision of the underdog mathmatical hero , misunderstood by his generation has been dispelled by his own words. As a free thinker at a truly unique time in world history, Hermann Grassmann saw something buried that no one else paid ny attention to, nor valued.  His hard working spirit and youthful curiosity kept him digging at it until he was drawn into a realisation that he was onto something!

He required the technology of his day to further extract what he was uncovering. That meant that as he matured he had to learn how to use various bits of philosophical and mathmatical machinery , so he had to take courses, read books, self actuated learning, self reliant study, self directed research finding out what he needed to know and do to uncover what he had Lund and evaluate the worth of it to his society.

It gradually dawned on him that in his extensive self education nd skill learning programme as well as his socil nd cultural development, his spiritual and philosophical advancement no one had expressed or revealed what was now revealing itself to him! So he was now hooked, convinced he was receiving divine inspiration or at least Muse lead insight and gifts.

What he saw were several fundamental entities that lived and breathed at the very heart and centre of all human experience of the geometrical Space, the Raum. These entities were real and complex, but constantly there in one form or another. They were of long long provenance, salled different things over history and glimpsed many ways but not distinctly uncovered.

The more Hetmann studied the more he realised how huge his discoveries were, and that how privileged he was, because the greatest minds of his era, and arguably of any era had not uncovered it.

It turns out that what he saw and described in utter detail was the matrix array.

We call it a matrix because of Cayley and his collaborators , but prior to that it was called simply LE Tableu , the table , in French. It was studied in the form of systems of equations , mostly linear , and the determinant, which I believe Cauchy and Gauss corresponded on. The determinant of a system was clearly a concpt of the times, and Hermann knew about it from the best minds of his age. But what they had not done was penetrate to the core ideas of these systems. Neither had they worked out how to systematically do this or what tools were needed to advance this development of thinking. There was no clear thinking about these systems that appeared everywhere!

Clear thinking for Hermann arrived in the form of Gottfried Wilhelm Hegel. He was the most famous philosopher of Hermanns era, and that is saying something! For Hermann who appears to have attended at least some of his lectures, or heard about them through his brother Robert who did go to Berlin to qualify as a Mathematical Teacher, something Hetmann struggled to achieve,  the consequence was that Hegel taught in his lectures by demonstration a powerful heuristic method and approach to philosophising about just snout anything. But in particular how the historical development of ideas, and deep deep concepts of life and cultures followed a pattern of advancement toward a higher and wider and fuller capability to reason and puzzle things out.

Thus Hegel powerfully linked every aspect of any individuals experince to this grounded real experience of history. Hetmnn lived in historically tumultuous times, life and death had to be faced o some kind of Readon led or religion led thinking.

Once grasped Hermann applied this meditative praxis of Hegel to everything fearlessly. You have one life and it should be a good one , delivering something of import. Hegels method or praxis of meditation contrasts with Descartes, who spent half the morning engaged in contemplative reflection, and the rest of the day going about his other duties. Hegel on the other hand lectured profusely, prepared for lectures assiduously, and ade lecture notes copiously and haphazardly, all so that when he was in front of the students he could perform! And what a performance! Intriguing, spectacular, stuttering and full of pregnant pauses, false starts, reiteration , upon reiteration until the connection and ideas were clearly seen by him, and then presented in such a breath takinly easy flow of pure philosophical gold.

Hardworking and deep ancestor linking and metaphor making  always lead to some advancement, Hetmann learned. This is precisely what he used to develop his discovery of the connection between points and line segments as formal concepts and reference points and kinematics in 3d space.

The 1844 book was intended to be a guide to a meditative praxis which would help to work through and refine thereby, the difficulties that the matrix array was ensnared in at the then proto stage of development. Hetmann did not say in 1844 he had the solution, he just claimed he had the method that would deliver he solution!

Of course we know now that Caley and others worked hard to develop the kinds of algebraic labels Hermann envisaged and constantly discussed in 1844.

The 1862 book is a different work all together, highly because it was redacted by Robert , Hermanns brother , with Hetmanns reluctant cooperation, and aimed at Mathmaticians rather than t classics students , who would recognise the references to philosophical concepts and philosophers underpinning the 1844 version.

Because Robert, as an experienced publisher, was right about the need to redact the work for a different audience, as proved by the second versions steady and important sales, Hetmnn was not going to be ungracious and criticise it as betraying his ideas, rather he utilised the success to reprint his original with additional addendums in 1877 etc. and added copious footnotes to connect the 2.

[jehovajah]

Ausdehnunglehre 1844

Induction

16. Therehere is the expertise-like presentation to its apparent nature/ character concording to a " gripping in one another " entity of two developing rank arrays, from which arrays entity the one with consequence drives from one truth to the other , and builds a representation of the central- like contents, but the other by governing the everyway travelling itself and the thought pattern appointing
In the mathematic these two developing arrays step at the most sharpened out of one another.

There has long time already in the mathematic, and Euclid himself has given the prototype therein, been conventions, only the one developing array,  which builds a model of the contral-like content ,  to allow to step hereforward, in relating onto the other thing, but it is left up to the reader it between the lines to read here out. Also Alone how  fulfilled the side by side ordering and presentation of of that developing rank array   is to be :thus is it there yet impossible, therethrough that array ,  of which the expertise should first Learn to know, already on every   points entity of the developing the overview is waiting against to support, and to set it on the bench, independent and free further to advance.

Thereto is much more necessary, that that the reader makes possible that in that array comes to be set condition/status marker every possible way   in which marker the one embedding  cover of the truth  should himself determine   the most favourable  case .

Therefore in  that array entity, which the truth uncovers, one continuous entity finds  place to sense itself over the journey of the developing.;
  It builds a copy of itself in the array to it, a centred property-like thought array
    over the way,  which it has knocked into one,
    and over the idea, which  lies at the foundation to the whole entity

and this thought array copies a central kernel/core and spirit of its Activity, while the consequential "Dividing apart" of the truths is only the everyway bodying of that idea .

Now   wishing to emote the reader, that he,  without being guided toward such thought arrays,  who still independently should advance onto the way entity of the embedded covering,
calling him  to place himself over the one embedding the cover of the Truth , and therewith is the proportional arrangement  between him and the writer ( pen man) to switch round,
whereby then the whole " away from writing planning" of the work appears as superfluous.  

Therehere have we new Mathematicians, and particularly to begin with the French, weaving together in everyway both developing arrays.

The "magnetic" entity, what therethrough  having brought  their works on the scene, plainly remains therin, that the reader himself feels free and is not hemmed in  in  thought patterns, in which patterns , because he is not mastering them, he must follow like a minion.

Now that in the mathematic these developing arrays at the sharpened edge step  out from one other:  lying in the central property-like quality of their methods (no.13); there it specifically from the specific entity out through chain linking advances, the Monad of the idea thus is the last entity!

Therehere slowly drags along with itself the second developing array a completely "rooted opposingly  set" character , how the first entity ; and the inter penetration of both appears more difficult , how in some "random" from a different expertise.

Therefore for the Sake of this  difficulty one may not allow, how it from the German mathematicians frequently happens,  the whole everyway procedure to give up on, and to throw out in every direction !

In the laid before you published  work, I have therehere the applied indicated duty Way knocked into one,

and it shone around to me, this work by considering a new expertise,  as needful ,

as plainly at the same time the Idea of the same expertise should step initially out of Light !

[jehovajah]

Commentary

For some reason this has been a subtle translation with a lot going on in the ideas behind the German, so I would very much appreciate comments on how understandable this section is.

As it is the end of the induction it is necessarily difficult, because the analogy of being given birth to from the Matrix, the Womb, is subconsciously and analogically driving his description here. As you may imagine this affects the reader at many levels and I guess as translator it effects me more. Being in the womb is such a safe secure place with all your basic needs provided for, so the trauma of being birthed or ejected is real, and Hermann means to eject us into the real world which he has prepared us for!

Well I guess I should be excited to face the challenges he refers to, but right now I just want to stay in the womb just a little bit longer!

However this womb I can revisit as often as I like, to be rebirthed, so that is encouraging.

But what of the matrices and arrays we are now equipped to look for and develop? Well certainly Norman has been acting as a good Midwife all along, so at least we can hide under his wings for a little while longer as we grow in ability and confidence. And it is clear that if you were and are German/ Prussian Hermann wants you to do just that, to take your place in the World as a new kind of Mathematician, first equalling and then by God exceeding the French! And this Ausdehnungdlehre is the work set before you to stick you to your duties!

More than that this work shines out from the Light, it is a work of inspiration from the Holy Ghost, the divine Reason that shone upon the shepherds and set them on their journey to the infant Jesus. And this divine or Absolute Reason is what Hegel was referring to as the advancement achieved through the dialectical process. Here Hermann claims the Rank Matrix is an embodiment of that truth, that reality of Space that is at its core, and there is a second like unto it which was the more difficult to integrate, but which it was his duty and the new Mathematicians duty to achieve!

[jehovajah]

There are several key points I missed coming at Grassmann the way I did, as an enquiry about the mess I perceived as the vector product and the complex numbers.

Much has been straightened out because there was much I simply did not apprehend, even from Normn. However I started thinking that the line segment was the fundamental of Grassmanns method and idea. I have ended realising that a much more complex entity even than Tensors was what he perceived: the Matrix.

On the way I realised I had not been aware of the potential point, the counterpoint to the line segment, nor the application of extensive and intensive magnitude. Further I had not grasped the rsnk array until quite late in the sections and had not understood the structure of the systems he revealed before me.

Then he mentions the 2 types of systems covering the same step rank at the second step. Or stage system, and finally how the  n- stage system comes to be.indeed how anything comes to be is addressed! But then he goes on to an entity I barely could translate until I knew the difference the adjectival endings make: the points Entity of the developing Matrix.

What that was or is I do not yet know, but I know it is to be found in the works of LaGrange and LaPlace. The formal model of this matrix is the thought Matrix and how that differs from the LaPlace and LaGrange Matrices I do not know, or even if it is different from the second developing Matrix which posed such a difficulty at the time of his writing.

Clearly Points and Point entities suoerscede the line segments as one goes higher in the system stages but how I have no clue at this moment.

The other important message was the necessity of an overview! That and a woefully lacking rigour were 2 skill set attributes he mentioned. Given that point the next chapter or essay deals with the overview of the doctrine of the thought patterns. Whether I need it to continue with this threads aim I shall determine by and by..as at this moment I need hermanns definition of the stepping out product and some indication of his roots of unity product to be ready to translate this paper on the place of the Hamiltonian quaternions in the doctrine of extensive/ extending magnitude.

[jehovajah]

On page 64 of the 1844 Ausdehnungslehre Hermann gives an anology to convince the reader that the out stepping product is anti commutative. It is an analogy because he is wanting to establish a principle for the outstepping product . Up until then Hermann has given many pages of demonstrations of the consistency of the result from the perspective of his defined 2nd rank system.

What Hermann has done is simple and clear, once you clear your mind of other thought patterns and in particular arithmetic of numbers. He has returned to the Arithmoi and the methods for handling geometrical extensions by them. I say returned because arithmetic of numbers grew out of this spatial handling of Arithmoi. This may not be precisely how the Greeks did it but it is very close.

As Hermann advised the reader needs to read between the lines here and then read the original Stoikeia away from this work. Comparing the 2 should give the reader a self established view on the matter.

So why the analogy, which forevermore has been taken incorrectly to define the outer product? It may well define the outer product  ofab as absinø the angle between the line segment but it does not define the outstripping product of Hermanns 2nd step system.

In the second step/ rank system every line segment is referenced by 2 elemental line segments in a prescribed process. In the analogy Hermann illustrates how such a prescription of process , when rigorously applied leads to the exposited result. It is the lack of rigour , the sloppy conventional thought pattern that gives us commutativity without demonstration. In his system ab has a as the principal orientation. Thus any other orientation is defined as measured from a to b. Thus ba reverses the principal orientation. Supposing now the line segments remain the same in space( or in the plane)  because they are bound to real points then ba can only be constructed by going backwards along a. This is reflected in the way the orientation of a is defined: it is measured fromb

The reversal of the way the angle is measured is an unexpected contributor to the sign change! If we are strict we cannot avoid this result, nor should we. It tells us that we are doing way more complex things in our processe than we allow for or even recognise. The protestation that we have to simplify things else we become confused, oes not wash with Hermann. There is simple and there is sloppy!

The call for rigour is thus well demonstrated in the way his whole system of systems is set up.

Now I have used the notion of cyclical rotation to grap this point as simply as I could.mthe point about cyclical interchange is the points! As the point labels move round the or imitations and principal directions do. Not follow. Labels are formal identifiers, but orientation and principal directions are spatial/geometric realities, set as such by our own thought pattern that is granted, but set in real space nonetheless and not formal space. So I may identify real points in space to do this, and once thst is done they are set forever for all intents and purposes.

Thus on earth the north and south poles are such real set points. To be flipping real poits around is to create mental disconnect with the spatial environment we exist in, and thus to lead to confusion of mind.

So the same 2nd stage system can also record formal rotation by label interchange or real rotation by point motion in and through a fixed subsystem of real reference points. This is the important realisation: to record rotation of a point or any motion of a point we require a 4th stage system, or higher!

The way it works is by imposing rigid rules on the subsystemat as a Spaciometry and then rigid or flexible rules on a local system representing the moving body. The spaciometric and the local systems thus combine in a higher rank system to describe dynamic motion.

But there is a systematic assumption that is not rigorous in most minds and that is regarding the role of Shunya or the concept of everything!

[jehovajah]

I have written quite extensively about Shunya since I fell into its warm limpid pools during my journey in the fractal foundations thread.( google jehovajah Shunya).

Suffice it to say that Shunya means everything. It was misunderstood from the day Brahmagupta formalised the astrological concept that it is, and has created panic and dread from its inception due to Brahmagupta terming the tally marks he uncovered " misfortunes"! And they continue to psychologically cause dread and loathing today as negative numbers, and most assuredly have caused psychological confusion in the hunt for the meaning of !

However Shunya was brutally tamed by the Arabs who started the chain from Shunya meaning everything, to Sifre meaning wind which eventually became zero meaning nothing to be afraid of! Then thoroughly whipped it was assigned its duties as place holder!  However in the real world it cannot be so easily tamed. It retains its infinite power. Truly of all the mathematicians Euler is the one who explored this best in the west, driven by his intrigue of Indian mystical mathematics.

While cantor was going mad establishing his infinite set theory Euler had already used circles that intersect as diagrams to apprehend how sets come to be out of Shunya. They are mistakenly called Venn Diagrams, but before that they were called Eulers rings.

In addition Euler used his rings to picture the great Indian cycles of ages or Yugas, and he also used them to partition the collections of counting numbers. By so doing he explored the Indian Decimal system based on these cycles great and small and lead the field in modulo arithmetics and equivalence Class studies. All of this derives from his intuition that Shunya meant infinity.
http://baharna.com/karma/yuga.htm

As a consequence he came to name i for infinite magnitude but came to associate it with , through his exploration of the trig ratios of the unit circle , but also of the conic section curves. All this he brought to our attention in his inimitable clear style, revealing what Newton had kept as secrets from the world.

Thus Shunya when placed in its rightful place that is according to its astrological meaning cannot be denied as a concept of infinitude.  But Brahmagupta did not mean to say men can apprehend infinite things and infinite processes, rather he meant and does mean infinity starts and ends precisely at each individuals location, that infinity is relative and we can only grasp it as the limbs of great unfathomsble circles that start at us, go off in one direction and return from the opposite direction.

The great Indian time cycles are astrological, and if you are moving through them and increasing in fortune then what was prior to now is relatively speaking misfortune. To move backwards in these great cycles is thus to enter into misfortune. Where you are at this moment in time is the balance between fortune and misfortune.

Vice versa a man moving into misfortune in these great cycles has indeed a great deal to worry about, for how is he to reverse his fortunes, except by the Gods? The hope was that Astrologers could alleviate the situation by pointing out the Kairos, the opportune time to jump ship onto a more fortunate cycle, when and if possible.

In Indian astrology each caste was assigned its particular heavenly cycle , and since the cycles rarely changed the caste rarely changed its assigned status. Everybody was happy with that because it was in accord with the will of Brahma, and it was not a constraint on fortune or misfortune. It was a divine organisation of society concording to the heavens.

So where Shunya in the Sanskrit means swollen, pregnant and ready to deliver the ancient Indian philosophy took that to mean full, and thus time to fill a new container. Using this system of full containers they constructed the modulo 10 Arithmos as power series containers of ever increasing magnitude. The symbol for full is o. This came to be 0 in the Arabic numerals and was placed at the beginning not the end of a cycle! Geometrically it hardly matters but psychologically it reduces our understanding of the origin of all extensive magnitudes.

All extensive magnitudes come out of the origin, thus thst point must have a super potential! In fact it has an infinite potential. When we ground a formal Grassmann system on a real point we are imposing a super potential formally on that point. However, because of the way the system within systems is constructed the origin of the 1st rank system is at a different potential to the origin of the 2nd rank system, and so on through all the stages or steps of the systems. What this effectively means is that within the origin of a 2nd rank systemat a whole first rank system is contained as a potential, and indeed is mistakenly set to zero rather than to Shunya!

In the real world we cannot set the points of an nth- stage system to Absolute 0. The potential of the points drops down to the previous level, and unless there is a complete collapse to all levels or stages, remains at the level where it maintains some potential.

So the anti commutativity is a relativistic one and applies only at the level of Shunya where it is identified. At all lower levels there will be tail ends that may substantially alter the real and empirically observed behaviour of a phenomena modelled by a Grassmann system, especially if it is dynamic.

Shunya IS Everything!

[jehovajah]

In the case of the out stepping product the system process is the same For the sum. The principal system varies first then the second elemental within it varies second . Thus this coupled combination produces the diagonal labelled by the sum rule and the parallelogram fletch which is identified by the product rule. The labelling is just identifying the aspects which result from the process of stepping out.

Thus we can expect a different focus from the colliding product .as far as I can determine that process is casting a shadow, or projecting a shadow .

The out stepping product is therefore naturally physically dynamic and relates well to mechanical systems of rigid bodies. Or coupled forces.

The second colliding product which later becomes termed thus is physically about casting shadows or projection and thus dynamically relates to light and electromagnetics especially reflection, refraction and diffraction., as assumed to travel in straight lines. The product is a line segment or projected image/ shadow.

[jehovajah]

Stormin' Norman!

<a href="http://www.youtube.com/v/rCDRCGjmaO8&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/rCDRCGjmaO8&rel=1&fs=1&hd=1</a>
<a href="http://www.youtube.com/v/rCDRCGjmaO8&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/rCDRCGjmaO8&rel=1&fs=1&hd=1</a>

Of course we part company at numbers!

Oh and also on the postulates( demands ) of Euclid.

[jehovajah]

After extensive scanning of the main text of the 1844 versioning have determined that I indeed do need to translate the Übersicht.

The induction , adds to the Vorrede , by design, because Hermann leads you to the importance of the induction in the Vorrede. And likewise it cannot escape notice the important role he places on the Übersicht in the Induction. I have translated the term as Overview.

However further there is the importance of the Hegelian triad , which would necessitate the Overview as bring an important part in the dialectic process.

Now Hermann bemoaned the imperfection of the 1844 version and I can concur that after scanning hundreds of pages and coming upon many surprises I was left dissatisfied at the end. The book falls short of its great potential! Some may say it still is sublimely effective, and I would concur, but it , judged by its own proclamations does not deliver that rounded wholeness expected from the Vorrede..

Indeed it was prockaimed that a second book would be necessary to complete the work, and because I know such a work was scrapped by Robert , and a different volume was masterminded by him I feel that we have missed out on an original Tour de Force.

The 1862 version is a different book, much less adherent to the philosophical and religious and mystical scheme that drove Hermann in the 1844 version . Thus hermanns copious notes and annotations and addenda are a belated attempt to unify the 2 thematically.

Hermann simply did not have time left to write the second volume as he had wished. He suffered from a kind of consumptive hay fever most of his life, he had I think 10 or so children and a wife to support and he had demanding duties as a linguistics professor working on the greatest work of linguistics ever devised, the Proto Indo European language lexicons and dictionaries. And he had a few ideas that were of importance in physics to do with the colour spectrum nd photography, as well as electromagnetism.

Had he lived longer I think he would have written his second volume as planned.

Some surprises are: that the stepping out product has an inverse, or a division to its multiplication!  As he developed the labels it became clear that the symbol was a commutative operator, unlike the stepping out product itself. This is not surprising when you drop the Maths speak and comsider the processes directly.

Extending a magnitude goes only in one Continuous direction, and summation is continuous/ contiguous.

However division is not an extensive process but a disaggregative one. Thus establishing the process by the label means that in using the label in a product one is not using a the product to multiply, but rather using it to divide, so for example the product 1/2 x 3 is a division written as a product , consequently 3 x 1/2 means precisely the same division.

How we change our view between the 2 kinds of multiplication requires a full reading, but essential the difference is in one extensive magnitude being out stepping while the other is being in stepping/ inversive. 

If the reciprocal were in fact a compression of a volumetric magnitude, we might refer to it as an intensive magnitude, but as the reduction cours as a disaggregation of the volume we can only refer to it as in stepping. Reciprocal and inversive are more mathematical terms.

How Hermann then goes on to establish the shadow casting and projection product is quite amazing, but his secret is to establish very secure algebraic labels and useage, and then to take those labels and "extending" their applicability to increasingly wider and unexpected fields.

Amongst these is the solution of systems of Equations, for which he writes down a solution in a one line Combinatorially variant set of coefficients for the system of equations. This one line answer is in fact the determinant expanded for n-equations of n unknowns, thus a nth stage system. For these equations he introduces the term Grade.

There is much more , but nothing on the circle, I thought until I understood the term Richtsystem to not refer to a Rechts angle coordinate system, but rather to a Richtung system, or in other words an orientation system, where translation is sintered out leaving just the rotationl orientation .

It was also noteable that Hermmann refers to the number count of the extensive magnitudes, not to the number per se! I related this to the Arithmoi, and indeed Hermann uses the tern Zahlengrösse which literally means tally count magnitudes and which I immediately understood as Arithmoi, that is unit magnitudes/ Metrons used to count out space. These unit magnitudes are the Euclidesn monads..

Then inexplicably Hermann starts a section called the Elementargroesse. In this he basically seems to repeat the first section called the Ausdehnungsgroesse. I was surprised, and that is why I think the overview is essential reading( as well as some sections in the Ausdehnungsgröße itself) .

Essentially think this is not to be translated elementary magnitudes, but rather elemental magnitudes of the first, second, ... n-th rank system, in other words what magnitudes make up the basis for any system, and how those magnitudes are coupled together to form an elemental magnitude for the system. Thus for lines the elementals are points for planes the elementals are lines, for space the elementals are planes, and for 4 dimensional systems the elementals are spaces.

This relates to some confusing ideas in the Clifford algebras where the outer product is used to define for example a bivector. Of course this is just a plane , but it is the elemental of a Raum or 3d space .

Now the inner product is a projected line segment , and being a line segment is the element of the plane or 2 d space. However when a line and a plane are composed in a sum , that is they are formed into a combination that represents a basis system , at least as Grassmanns colliding product composed with Grassmans outstripping product the form a basis of 3 elemental magnitudes. As such they are the elemental of a 4 dimensional space or a 4th rank system.

So why not just go straight to the 3 element basis ?

Well firstly because Hamilton, not Hermann had clearly shown the need for a 4 dimensionl basis to do rotations. Hermann therefore is responding to this fact not by claiming he had discovered it, but rather by saying, now it has been discovered let us analyse it and synthesise it by the methods of the Extending/ extensive magnitudes.

Secondly he is able to draw on the rich work he has done regarding the projective product and the cyclic interchange in the plane to explain the anti commutativity of Hamiltons quaternions.

Thirdly he can show clearly how the same extensive magnitudes characterise the rotation  by giving the point of rotation and the plane of rotation.

To understand this Normans excellent series on the quaternion shows how a composition of rotations ends up being described by just one Ritation , and in Hermans ideas each rotation has a fixed plane in which it swings , thus we need to define the plane of swing by the start line on the edge of the plane and the  finish line , by which we then can draw a parallelogram plane encapsulating the composed rotation.

Of course when you are used to rotating around axes this is difficult to conceive, but Norman uses reflection in a line or plane to compose rotation. He could equally have used just projection. Had he done so he would in fact have explained the essence of Grassmann rotation and representation of the quaternions.

[jehovajah]

I would not want the reader to go away without appreciating the complete mundanity of Hermanns vision, how in fact these truths are everyway bodied( consubstantiated) in the objects and processes around in the world. For example I can hardly move without recognising some array , or rank array around me. Particularly when I go into a department store I see row upon row of arrays  stacked with items for sale, or enticingly or ergonomic ally designed stands offering up or setting out what the shop has to sell. The department store itself as a building is a complex array , but the important point is not to get stuck with the supposedly mathematical concept of an array.

To begin to liberate your mind I recommend looking over the work of Escher, who more than most captured the essentials of what his contemporaries were conceiving, but in particular I think gives aesthetic form to Hermans concepts of the thought array, and the developing Array , and also the points entity of the developing array.


MC Escher

Consequently we are led to address the curious way we attempt to understand 3 d space. Some of us apprehend it through the great works of Sculptors and Artists and Artisans as Architects both civil and military . The Stoa in Athens is architecture that gave its name and form to a school of philosophy , the Stoics. The Peripatetics were an architectural feature frequented by Aristotle and his Students at the Lyceum and gave their name to his students and teachers who spread throughout the Hellenistic nd Arabic empires.

Art in the form of Mosaics is a ground breaking fundamental symbolic mystery or secret of the Pythagoreans, whose mystical Gematria has been misunderstood and misrepresented as mystical nonsense, whereas they are actual forms in space , standardised by the term Arithmos, made up of units not necessarily uniform tiles!

One of the great obfuscations of religious and cultural translators is to take the mundane and make it mythical or legendary. Few would wax lyrical about an abacus, and yet it is one of the fundamental calculating arrays used in the ancient Chinese and Jaoanese cultures. Many have waxed lyrical about the Indian decimal system, but they have stripped away the Indian temple and deistic mythology which helped to preserve the information in a oral tradition passed down for millennia as Ganitas or lines of instruction and construction, governed by the great Gunas identified by Indian scholars of great sophistication, and the Astrologers who devised an array of cyclical clocks known as the Yugas and other huge cycles.

These are all arrays embedded in memorable mythologies to aid the transmission by oral means of much empirical expertise.

The Kaballah cannot go unmentioned. For centuries mystified as some ancient dark magical system the Kabballah is merely the accounting and calculating systems the Arabs gathered together from all the cultures in their vast empire. But since, as all these things were, it is based on actual spatial Metrons not symbolic numerals or concepts of number divorced from space, these systems embed in a Spaciometry and are shot through with spaciometric insights. The concept of number divorces the modern mind from this and  karals it into a sterile and death dealing place. Mathematics based on that is a dead thing, soon to rot into dust and blow away in the winds of advancing history.( that is, the things to come viewed as if they had happened in the past!).

So now, we have made strenuous efforts to understand these arrays in 3 d and 2 d, that is as drawings or notation on a page. And since Regiomantus the sprawling print versions, being costly have had to be abridged somehow and it has been accepted that the printer has the final word on that . Thus we get square regimented blocks of typeface as the standard and most economical way to represent what is a living reality.

The square or rectangular arrays arose from a combination of printing convenience, cost, full utilisation of expensive materials and conciseness. However they have now constrained the thinking of many generations in a way our forebears would not recognise! A tree is an array, a cellular organism is an array , a shattered plate is an array! That is why I use the term mosaic advisedly, and the thought array may well get translated as the thought Mosaic, which is formed by apprehending and populating these arrays as formal structures with empirical truths.

Where we have been taught to fill an array with numbers as data points or measurements, Hermann conceived them as status or condition markers encapsulating the truth of a condition at a moment. The developing array allows that identified array to dynamically change and then be captured at a final moment. It is the relationship between these 2 array instances that underpins the LaGrangian method, and also develops the thought array in the reader as a dynamically developing thing.

These are or were hugely difficult arrangements to calculate, despite the notions being or becoming intuitively obvious. Between the initial and final states of the developing array all that was conceivable was that points moved relative to one another. How they moved could not ultimately be determined empirically, but certain constraints could be imposed. It therefore became important to analyse the correct constraints to see if they produced the final array by a calculation process alone.

Using this concept snd the calculus of variations idea LaGrance was able to potentially describe any dynamic system by a calculated process. To do this he of course adopted the laziest assumptions. I mean why work too hard at a complicated system? But to his amazement this lazy approach more often gave him the right answers within bounds.

Hamilton picking up on this redefined Mechanics using the Hamiltonian array as evoking the principle of least action. It is still  a standard approach today, although both Euler and LaGrange agreed that it was not a foundational principle of mechanics like the system set up by Newton.

And so we come to Einstein, who really did not have a mathematical clue how to compose his insights. His insights and instincts however are the crucial thing, that is his thought array or thought Mosaic is what has been used to apprehend rank arrays in space and to populate them with empirical measurements and then to calculate the outcomes. His type and style of thought mosaic/ array became known as special and general relativity, and was meant to be a more fundmental setting for geometry in which to apply Newtonian Mechanics and astrological principles.

Suffice it to say here that Einstein as well as being a catalyst for a brave new geometrical or spaciometric approach, was alo a political pawn in a propaganda game that existed throughout the second world war and well into the cold war period. Much of what is set out about him was myth. A lot of his ideas were taken from colleagues in Europe who were unable to publish because of the political climate, and so he published on their behalf. However it was not possible to use certain words, particularly aether, because the security services used that to identify actual enemy spies! Thus the notion of spacetime, taken from Hamiltons work was born. Spacetime has always been a mathematical or geometrical model of a rotational aether.

Einstein, later in life , when the climate was again safer gave lectures in America and Europe calling for the return of the Aether to the praxis of Physics.

As usual I originally started this post with the observation that the complex plane, the vector plane and the hyperbolic plane are just 2 dimensional slices from  a Grassmann n-th rank space or system, and therefore what we have called " imaginary" are in fact just useful parameters for thought patterns that describe the field of study.

Thus we only have translation, rotation and projection in a Grassmann system and it must be one of these that must account for the strange behaviour we first identified as . That being the case means that we can identify the other formal strange behaviours we may encounter by findng the rank of a system in which that behaviour is quite usual! The reduction or projection onto a plane will thus account for its mysterious behaviours( according to our planar point of view).

[jehovajah]

I forgot to add how the rise of video and high speed photography and filming has made much of the old calculative methods really obsolete. We do need to shift paradigms to a more fluid rendition of these developing arrays and the point entities associated with them.

Differential calculus can now be replaced or refined by high speed filming, each frame is a developing array filled with the truth. By the end of the film we can see how the point entity in the array has changed. We may not have a formula to differentiate to concur with what we can now plainly see with our own eyes frame by frame, but we can use the video data to establish the most fitting constraints on a model system  to mimic the filmed behaviour.

We now have the ability to check our physical laws and algorithms against instantaneous( almost) data sets. Instead we still assert our physical laws are correct and natural behaviours should obey them!

[jehovajah]

I cannot pass on without drawing attention to the fractal generator.

These applications are natural developments of the use of developing arrays based on poit entities . We know that interest in iterative systems of Equations, or even synthetic geometrical iterations existed before Benoit Mandelbrot. In fact he drew on a french research interest in second world war Europe in formulating his ideas. Work by Argand, Fatou and others .In America little known geometers were also researching the effects of iterating geometrical forms at different scales.

It was the arrival of very powerful iterators that made possible the formulation of Mandrlbrot's ideas. But it required the development of graphical plotters to bring fractal geometry to light and life. The discovery of the Mandelbrot set was to be the start of a whole new appreciation of the point entities and the developing arrays.



The use of colour coding and colour cycling added a new dimension to the imaging of these sets, and the creation or rather recreation of the geometry that first developed the arrays in LaGrange and LaPlaces mind, the thought arrays. However now it was possible , through the developing array to establish point entities and to constrain their development by simple rules over and above the elemental rules of the reference frame system itself. By colour coding certain point displacements it became clear that there was a central core/ kernel as Grassmann said , that characterised the whole system. The most famous iconic image of the Mandelbrot set in the complex plane was born.

These concepts were hard to explain to ordinary mathematicians. They had no clue or conception of what Mandrlbrot was talking about nor what Fractal Grometry even was. It required the development of new mathematicians, just as Hermann had said, engineers nd physicists who required powerful iterative techniques to solve real problems described in array format. These array systems solved linear equations for all sorts of construction and manufacturing issues, but it was the computer generated image artists who made the breakthrough Mandelbrot seemed to be describing!

Suddenly instead of numbers in endless arrays being printed, images of lines and triangles and colours were being printed. Besides the Mandrlbrot se icon that Brnoit published showing the symmetry of certain complex planar sets under simple rules. There was little else until an aeronautics engineer applied benoits methods to generate realistic mountainous terrains from iterated triangles!

Slowly but surely the geometry of the real world was revealing itself through the transformation of these point entities. According to simple scaled rules iterated a number of times to develop the arrays, and then plotted as line segments and 2 dimensional forms, then surface coloured cvordingly to ome other simple rule, or colouring array.

The history of fractal geometries until now is fascinating. While still alive Brnoit tried to spread the word about the importance of his discovery and the new field he was researching. It is fair to say that the old guard I'd not take to kindly to his brand of mathematics and self promotion, but today largely due to his efforts a new mathematical cadre is well versed in the fractal geometrical point of view. What they lack is the Hermann Grassmann method to make sound sense of it all.

Today many enthusiasts hang on to the subject by immersing themselves in a practical fractal generator way, but the complex math behind it they would find hard to understand because mathematicians do not understand it themselves! They do not understand because they often distance themselves from coding algorithms and programming. Computer scientists do understand how the programme works but they throw up their hands at the " math"!  The left hand does not know what the right hand is doing.

Fortunately, Hermann Grassmann set out a way to explain all of this in really simple terms. But it does require the reader to read the Foreword, the Induction and the Overview of the doctrine of Extending/ extensive magnitudes, and to be willing to think differently..

[jehovajah]

As I retread sections of the paper regarding Hamiltons quaternions it is clear that my thought patterns have been well inducted by the translation exercise. Not only can I see what Hermann is saying more clearly in German, but also I now have anticipations of what he should BR expounding upon, and this guides the nuanced meaning of unfamiliar words..

I will next translate the overview but must once again reiterate the harm to your imagination that the number line concept is doing.

From the outset you can feel the dynamic energy of Hermanns Spaciometry. Nothing is static, all are thrusting or jostling magnitudes! Thus counting is a necessary method of kerping track on live entities. Where the entities are and how they move relative to one another is the experience of space and time within which we dance and sing a ritualistic count , or sequence. That sequence helps us to recall the Spaciometry and the dynamics of it. . Without that indissoluble link counting and numbers die and become poisonous. With the link fully recognised then the whole system is a light and sun filled day of joyous activity!it is of interest to me that Hermann refers to Helmholtz when discussing issues about the foundations of Geometry. In all my reading do far I have only come across one mention of Gauss by Hermann and none yet of Riemann. What that may mean I can only speculate on, but certainly Hermann looked to the French Ecole as the way forward on 1844, the standard to emulate and surpass. In 1877 he  reiterates that the grounding element of geometry for the Extensive magnitude doctrine is the line segment. This is because conceptually there is a duality between line and point that is indissoluble visually. Because geometry is perceived as a visual study field this link can not be ignored, without generating harmful effects.

However Spaciometry is in fact not a visual field of study. Hermanns system allows us to discover this experience by anchoring a reference frame to real points and then exploring the role of other senses.

One by one biological and neural science has uncovered the role of other sensors, particularly the proprioceptive ones in describing our 3d experience of space. The synaesthesia of these sensors explains why we often cannot distinguish a point from a line, but on a purely auditory sense points are the only perceptible intensities/ densities. Certainly orientation , the essential relative property of a poit to a listener is distinguishable, but any line or path connecting or associated to the orientation or change in orientation is not perceived auditorially . That line is the main attributable difference in visual processing over any other sensor processing.

Because Hermann takes care to include the thinker in the experience of the subject, hermanns system is so powerful and adaptable. Others have sought to remove the thinker in order to claim some " objective" reality.  For me that stance is fraught with srniry and psychological difficulty. It dehumanises our connection to our environment and creates artefacts as Ghosts in the machine. We never truly understand space that way, and it is only now that I realise we only ever approximate to a model of dynamic space by Hermanns method, but at least we do not feel alienated from our natural environment and intuitions,

[jehovajah]

You were born into a world where animates are known to count, and to visualise magnitudes, but no other compares to the human animate in the extent and intensity of that activity. So we humans count and we calculate.

Why do we calculate? Certainly as a result it gives us no more in numeral names than counting does. And that is the mindset that focussing solely on numeral and numbers engenders.

We calculate because it saves us time and effort in and through space. For the more astute we calculate because it enables us to travel through time and space at any scale with minimal effort. Calculation is our mental process of traversing a model of our reality to end up at a particular place in our model, with a particular aspect or view of that model, at a particular time in that view.

Not only do we calculate space and time but also energy and Dynmics. Calculation helps us to " move" without moving . To travelnastrally to any plane in our model and view and experience from that plane with all the assurance our model can give, but in less time and less space than punting would take.

When we count we should dance and sing, because the alternative is to trudge and mutter as we measure through out space one unit at a time.

When Napier discovered his logarithms it was through traversing around the perimeter of the broadest circle he could imagine. Logarithms were about movement and motion, but in such a way that each calculation determined a constrained motion on the arc! Of course " serious" mathematicians pooh poohed the idea preferring Bergis more " mathematical" series explanation. The dynamics of space were " killed" off again.

Berkeley weighed into Newtons Fluxions as mere religious figments, once again refusing to accept the dynamics of the Arithmoi, and eventually confusing Dedekind into a weird set concoction so unnatural and poisonous that we are dying from so called real numbers.. Berkley initiated a movement that lead to numbers and the number concept being divorced from reality, while ironically calling themselves Real Numbers. Arithmos and magnitude based on Monas / Metron now died away, to be replaced by ... Well nothing really. These numbers were supposed to be logically defensible, but in fact they were and are the feverish macinayions of an over sophisticated sensibility.

Returning to magnitude as experiential continua , either intensive or extensive, connects us back to space , time and proprioception of these and puts us or me right back at the centre of our/ my experiential continuum. From there I can make sense of all so called higher Maths

[jehovajah]

Normans <a href="http://www.youtube.com/v/uRKZnFAR7yw&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/uRKZnFAR7yw&rel=1&fs=1&hd=1</a>.
Note the double turn or arc length .

As Norman pointed out this is a basic circle geometric theorem. The double arc length subtended a single arc length on the diametric opposite to the centre. Otherwise stated: the angle subtended at the centre is 2 times the angle at the circumference on the same side of the chord!

While angles at the centre or corner are problematic, this is only because of misdirection. We measure angle on the arc not at the centre. In addition arc have always been associated to ratios of chords to diameters , so the argument about the trigonometric functions, for me is not substantial. The real issue is the real number concept .

However the rational treatment of trigonometry is a very good approach. The traditional trigonometric equations became so slippery in the function form that most physicists resorted to trig line segments to express these relationships.

Euler's theorem for the exponential expansion of the rotation of arc can be rewritten in terms of trig line segments, and immediately becomes an Ausdehnungsgröße, so a lot of the issues Norman has are catered for by Hermann.

Now the use of reflections arises naturally in this treatment by Norman, and leads to the double angle format. What I hope to show is that rotation can be described by projections, but the projections are "Backwards" so to speak. This then should link to the chord and arc ideas of Ptolemy, and the rhomboid plane of Hermann.