The Theory of Stretchy Thingys

[jehovajah]

Norman introduces the vector plane. This is a one to one correspondent with t is usually called the complex plane.
<a href="http://www.youtube.com/v/W9&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/W9&rel=1&fs=1&hd=1</a>

The vector plane removes the imaginary component completely and vectors simply do the job!
He goes on to define vector reflection.
<a href="http://www.youtube.com/v/HsQ0OSiAaow&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/HsQ0OSiAaow&rel=1&fs=1&hd=1</a>


Both videos highlight the simplification got by formalism and anti commutativity defined in the triangle product that mimics Pythagoras theorem .

What is missing in this development is the notion of a curve, later centre of curvature and arc lengths are reintroduced in the context of calculus. But I know curve is not fundamentally a product of calculus, rather calculus is a product of curve, as is continuity.

These are intuitively clear experiences that are hard to pin down. They are not meant to be defined by other easier concepts, although it is instructive to attempt to. They are fundamental prototypes that initiate new types of Arithmoi and schemes of arithmetic. Polar coordinates and Cartesian coordinates are an example of this point.

[jehovajah]

This overview of Grassmann thinking and difficulties really shows how a simple observation can lead to profound insight. The replacement of complex numbers by Strecke  is so fundamental that it is hard to apprehend why we do not start again, as Norman has done. In addition the fact that Strecke are different to points means that the points can remain fixed while the Strecke move ! The example of rotation is usually described as the whole plane moving. In fact it is a vector plane that lies coincident with every point that rotates relative to the fixed plane of points. And the vector plane can get screwed and twisted relative to the fixed point frame. This objectifies the vector plane, so that it is no longer referencing points in the reference frame, but vectors in its plane relate to each other as if the plane was some medium.

The vectors could be thought as attached to a rigid object . As the object moves relative to its initial position the vectors move with it.
If the medium is a fluid, as the fluid moves the vectors move with it .their relative positions describe the behaviour of the fluid.

[jehovajah]

At the heart of Grassmanns and Normans method of Analysis is an apprehension, conscious or otherwise , of the Logos  Analogos structure encapsulated by the mosaics , that is the Arithmoi.

I am going to explore in my blog the connections  between proportions , polynumbers and polynomials. I will provide a link when I am done.
http://my.opera.com/jehovajah/blog/2013/11/26/proportion-poly-numbers-and-polynomials


The theory of Ausdehnungs Größe is a method of studying forms. The Greeks did the same thing after the Pythagorean conception that space could be tiled in a mosaic. These were called Arithmoi. Using Arithmoi of all forms and complexions they could count space. We can count space.

But there is much more to the Arithmoi than counting space.

[jehovajah]

The polynumber theory in Normans math foundation are a fundmental shift to a logically consistent Algebra. It's background is explained at this link
http://my.opera.com/jehovajah/blog/2013/11/30/the-polynumbers

I initially recognised these polynumbers as Ausdehnungs Größe, but now I know they are not. The real difference is in anti commutativity. However everything else carries across between the 2 algebras. Arguably Norman's presentation is the better more fundamental exposition of both topics, and makes Clifford Algebras more accessible!
<a href="http://www.youtube.com/v/UMWbb-WAPsk&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/UMWbb-WAPsk&rel=1&fs=1&hd=1</a>

[jehovajah]

This and other videos in the series from video 55 actually, build the theory.
<a href="http://www.youtube.com/v/KbNYsNWRtzs&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/KbNYsNWRtzs&rel=1&fs=1&hd=1</a>

Norman makes a few mistakes and logical errors, but nothing serious, and in later videos he attempts to address the issue.

Of course I do not agree that he has corrected all of them, hence my blog post, but all that is needed is a clarification of the real origin of the ideas. In computing, the unseen element is the object oriented programming. This is as if it were a model of human subjective processing, and that is a breakthrough in apprehending what we are really doing with these things called " numbers"!

For this thread polynumbers are useful to grasp how Grassmann extended his arithmetic through the construction of the right triangle in notation, necessarily imposing anti commutativity to get what he prized: Pythagoras theorem. The consequence was that the commutative law in associative multiplication became " transitive" . In distributiyity the scalar is spread to the 2 elements of thr bracketed sum,
a(b+c)=ab+ac
But in associativity
A(BC)=(AB)C OR B(AC)

Ordinarily the OR would be an = sign but because of anti commutativity  the OR is replace by a =— ! This one change make polynumber arithmetic so called vector arithmetic.

Polynumbers in that regard found both matrix and vector algebras with the specified rule change. From this rule change the so called wedge product can be defined . This definition naturally relies on the combinatorial product of tables which can be cast in the role of matrices.

All these subtleties are tied together concretely and algorithmic ally in this computational Algrbra called the polynumbers.

[jehovajah]

<a href="http://www.youtube.com/v/Sbet5GhqmyI&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/Sbet5GhqmyI&rel=1&fs=1&hd=1</a>

This video was supposed to be the clincher. I have critiqued it in my blog. Nevertheless despite Norman finessing the fundamental computational structure behind this, I still would recommend it as indicative of the fundamental power of Normans approach to redefining" Algebra", and ultimately all of geometry and calculus.

The Trigonometry that introduces metric evaluation is not quite properly connected, but I will discuss how Grassmann discovered how to do this and why Clifford was so enamoured of Grassmanns method of Analysis.

[jehovajah]

The so called wedge product is a modern interpretation of a formalism that Hermann introduced when extending arithmetic to Dir Raume, or 3d space.

The product actually arises from the formalism of Justus Grassmann, and his attempt to place geometry, as exponded by Legendre on a logically consistent basis,  suitable for the primary education of the next generation of the Prussian intelligentsia. I won't go into the ramifications of the Humboldt reforms, but suffice it to say the Prussian Renaissance was pressingly overdue, and French expansionist goals had humiliated the once mighty Holy Romn Empire.

Hermann built on the Formalism of his dad. arithmetic was and is the greatest triumph of the Greek sciences and it was revered as a model of all systems of thought. Algebra was down graded by all but seen as a symbolic arithmetic that generalised certain results and methods allowing formulas nd algorithms to be expressed. Because of Descartes and DeFermat and Regiomontanus the metrication of other aspects of geometry was complete and formulae were in place for all kinds of situations. Consequently Synhetic geometry declined in importance academically placing it below algebra in most evaluations.

Justus therefore felt it was a suitable elementary subject for kids, if he could " iron out" a few logical inconsistencies! Further he could use it as a gateway to an understanding of Arithmetic and also logical thinking that up ports arithmetic.

However, his analysis ran into a subtle difficulty when it came to multiplication. In the end he found himself unsatisfactorily asserting it was obvious from geometry! Look at the geometry of the rectangle and multiplication and multiplication tables are an obvious consequence.

This may satisfy the casual reader, but it is logically tautological! And this bothered Justus a lot. It meant that the key operation in Arithmetic was fundamentally a geometric one, and not a logical or fomal concept. He had to have a prior geometry to define multiplication as it stood. It did not derive from the logical structures then taught in Prussian Higher institutes. No set theory could account for it , rather set theory required it to even exist in its higher forms.A x B where A and B are 2 sets is a clear mimicry of a geometric rectangular form, and Justus could not fathom it. It seemed one had to accept that logic was not the supreme source of all rational thought!

Hermann was thus used to a very very strict formalism which is why he noticed the letter patterns in the geometrical notation and the connection through the symbolic operators + and x.

The other aspect of the formalism was the growing, but still pioneering group theoretical approach to mathematical or rather algebraic analysis.

Hermanns law of 3 points hence the law of 2 Strecken  was the simplest summation form in geometry. 2 lines joined together. I all this the wedge join, just to confuse the issue! The term wedge here means to jam one hard into the other, and is not the same as the word wedge in the so called wedge product.

The summation of 2 Strecken in this specific way , precisely as defined by the order of the letter symbols for the points A,B,C allowed Hermann to later define a labelling identity

AC ~ AB +BC

While this looks simple and definite, once you allow a label to be an identity you can never stop!
So AB can be a label for 2 other Strecken and similarly  BC . The end result is a decomposition of the plane into nested  Strecken summations that sum to the above formula in some sense.

It is in defining that sense that Hermann developed his method of Strecken analysis, and the dynamic nature of Strecken.

The next defining moment was the realisation that Strecken multiply. This in itself is not earth shattering news, but in the formalism of his Father Hermann saw how this multiplication worked in a complex construction of a parallelogram. Unremarkably he could se in the labelling that  a fourth point was produced.
AB x AC = ABCD

This is what I called the fan product, and there was a wedge product version
AB x BC = ABCD

The fact that the D was in different relative positions was not lost on Hermann, but it did not seem to matter, until he combined the multiplication with a third Strecke in the plane. This was to demonstrate the associative rule in arithmetic.

AB x BC x EF. What did this produce?

Pretty soon Hermann was formulating how to construct this result using parallel lines. It is this formal construction tht underpinned his next discovery. By parallel line construction he could project any Strecke in the plane onto the original fan or wedge pair . Then, by the formal construction rules for parallelograms he could construct the various parallelograms in the product. Of course he ended up with a complex synthetic structure of overlapping parallelograms, but to do so he had to institute certain conventions. Using these conventions he attempted to combine parallelograms. It became apparent that these constructed forms would have to be signed if they were to cancel out in any way to substantiate the formulary.

When he moved on to the distributive rule everything started to fall into place and he saw a way to recover Pythagoras theorem from the notation alone. To do this he had to sacrifice commutativity at least in the triangle setting. That is AC x AC

When he did this identity expansion he realised that if he set AB x BC to —BC x AB  then he recovered the beautiful theorem.

At first he was shocked, because it seemed like he had to be wrong, but months of further work confirmed his decision. He not only worked out problems more simply using this rule, it also only made a difference in odd numbers of Strecken. Anti commutativity did not occur in the quadrilateral setting, but signed forms did..

So ther was no wedge product per se for Grassmann. There was just a realisation that Strecken multiplication was anti commutative to get a special result when a Strecke was produced with itself, and that Strecke could be constructed from other Strecke which could be set or defined as the basis Strecke.

The consequence of this was that AB could be arbitrarily used for numbers and Strecken multiplication. However the rules of the 2 types of magnitude had to be strictly adhered to, and not confused. Later mathematicians did confuse the 2 magnitudes and created some problems in interpretation.

These problems surface best when one looks at how Hermann came to the notion of " division".

[jehovajah]

The so called wedge product now has an interesting behaviour. In the triangle  or rather the law of 3 points
AB x BC can be multiplied by AC .

The construction is complex. What does it mean to construct a form from a parallelogram and a Strecke?
The consistent thing to do is to use parallel line projection. In the plane this results in overlapping forms , but in die Raume it results in a  parallelapioed if the third Strecke is not in the plane.

What one would ideally want is ( AB + BC)x AB x BC to construct the same forms as AC x AB x BC .

It is also consistent to define AB AB as 0 parallelogram. But one could also consider it as a degenerate parallelogram with lineal magnitude 2 AB .or – 2 AB ,

There are other combinations to define before the full import of this multiplication is understood.

[jehovajah]

As Hermann matured his ideas and notation he realised that the construction process was more simply identified as a lineal projection. That idea was in the air. Des Argues had pioneered a Grometry of perspective , with the aim of delivering realistic art into the 2 d frame. Now ponce let had revived interest in his work especially as it related to industrial and engineering design.

Hermann had entered a challenge to describe the ebb and flow of tides. He needed to pass this challenge in order to progress in his professional development. He read Laplace and Lagrange on celestial Mechanics. In it he came across the description of the half bow of a hyperbolic function. Much of it was written in unfamiliar mathematical,notation, but one idea shone out: the hyperbolic bow could be described by versions of the sin and cosine ratios. It was clear that these ratios , usually associated with angles and arcs were not tied to these measures. They were tied to the right triangle.

The idea was simple: model any curve by a collection of right triangles . Thus any curve could be tabulated like the sign and cosines by measuring or calculating the sine ratio for points on the curve. Equally by calculating the cosine ratio. In this case the ratios were eventually distinguishe by the nomenclature cosh and sinh.

Furthermore, the hypotenuse of the right triangle was a Strecke, and the sin and cos ratios were commonly seen as projections of this hypotenuse vertically( Senkrecht) or horizontally onto the right triangle sides. These projection lines were as much construction lines as parallel line projections , and later , perspective projections.

The nature of the product process was one truly about projection onto or into.


In passing, one projection remained unexplored and that is circular projection. This is where a point or a line is projected by a circle or a set of concentric circles.it is even possible to have a set of perspective circular projections where the circles are not concentric..

We continue with the direct or straight Strecke projections.

Two points A, B can form a product AB which is a projection by points joining A to B. it is either a line or a curve , the user defines its type.

Then we can define 2 lines AB, CD as forming a product that is a parallelogram. But they both are first projected onto a common point. This is called the meet, and then by parallel projection they project by each other to form the parallel sides of the parallelogram. By this means the parallelogram can be constructed anywhere in the plane.

With a third Strecke we can project by point, then by parallel lines. The Strecken act as projection directions before taking up their final position. Also. Parallel Strecken are sketched in where necessary to complete the form. The question is, How many are we allowed to sketch in? What is the final form?

[jehovajah]

So projection helped to clarify Hermanns thinking, especially about the product of Strecken.

He had made the fateful decision to let ab = – ba where a and b were Strecken acting as factors in a product. The product was constructed and it was defined as a parallelogram. The introduction of labelling and this Strecke sum or wedge combination led to a potential to recover the Pythagorean theorem, but he had to accept his new rule, mentioned before. Up until this point he had no division process and his method and Analysis was full of holes. He could use it to make incredible calculation streamlines and find general solutions, but only in specific cases, and in an as yet unorthodox manner. His arithmetic extention was not transferable to others.

The notion of a projection as a general construction process streamlined his understanding, and brought his ideas to an extended range of situations. He could now consider points as multiplying to project straight lines, and point sums he could properly define as the Schwerpunkt a point that counts connections to other points and Schwerpunkt.

The case of the right triangle was not general enough. It only really worked for rectangles.or Strecken at pi/2 radians to each other. There were 2 cosine laws that extended the Pythagoren theorem to general triangles . There was a direct connection to his decision to accept anti commutativity.

The term 2abcos( c ) resulted from the combinatorial product of adding a right triangle or subtracting a right tiangle to the general triangle to extend Pythagoras theorem to the side that is usually called the hypotenuse.in this case it was one of the Strecken determining the construction of a parallelogram. This triangle that was added to the general triangle was constructed by projection vertically. This is called in some texts dropping a perpendicular. This introduced the trigonometric ratios into the algebra and in so doing introduced ratios and ultimately division.

One important formalism that created this confused inight for Hermann: the horizontal layout on the page. AB x BC was laid out differently to BC x AB ! The first Strecke  was always the horizontal one! Consequently changing the order of the Strecken as factors changed the construction on the page. Subconsciously the constructions looked as if they were the negatives of each other in some way.

The projection representation of construction revealed that the real negative was associated with the trigonometric ratios and especially the cosine ratio. This removed some intransigent inconsistencies in his theory and Analysis thus far. The product could now be consistently associated to the trig ratios and the constructions guided by these ratios.  The anti commuting terms have to cancel, add positively or add negatively to be of most appropriate use, and this splits the method into 3 algebras. In one the anti commuting terms sum to 0, in the other 2 they sum to 2  or–2 or even 4 or –4. But the ratios provided a resolution which made the angle measurement unnecessary to resolve these decisions.

[jehovajah]

Hermann, in the law of 3 points or 2 Strecken recognised a summation of Strecken. In the law of 4 points or 3 Strecken he recognised the multiplication of Strecken and the distributive rule of multiplication, but also the rule of Strecken substitution or labelling. From this strong sentiment he actually concluded rashly that

ab = –ba.

This is the anti commutator , or at least its genesis. You might wonder why I say rashly, if you are used to the shock of this statement. Well that is because Hermann was deeply concerned by this conclusion. It put him off his dinner for several months!

Eventually he came to accept it, and this is why we have a so called vector calculus and algebra. However it is important not to get ahead of ourselves. Hemanns algebra differs from much of what is described as vector algebra, and that is because Hetmann never gave up on Strecken. In other words hemann was always trying to do synthetic constructive geometry in a more symbolic arithmetical way.

Later, hermnn came ino contact with LaGrange et al and projective Geometry. This refined nd corrected his views and lead him to redact his own ideas. The anti commutator came about When he realised that
If AC is set as the label for AB + BC then this sum can be substituted for AC in any multiplication.

In addition AC x AC is the product of 2 sums  or the binomial expansion of the substituted sums. This binomial expansion recovers the Pythgorean theorem precisely when AB x BC  = – BC x AB

But for the general triangle a more complicated version existed caled often the triangle coine law. This requires


AB x BC + BC x AC = 2 mn cos (ß)

Where ß is the metric of the angle at point B and m ,n are the metrics  of AB and BC .

This is the anticommutator in its general form written in Strecken notation.

Now you may be able to see how Grassmann intuitively made the correct decision .  When ß is pi/2 then AB x BC does equal –BC x AB! But otherwise it has the value of 2mn.cos(ß)!

The other trigonometric ratios also applied when projecting Strecken vertically( and horizontally).

The trig ratios might be thought to be exclusively about circle metrics, but his investigation of the hyperbolic function and curve opened his eyes to the more general, encodoing use of these ratios. We might think thar AC projected vertically onto another Strecke AB
 is AC cos(A) in the AB Strecke but that is not necessarily the case  it could be AC Cosh(æ) where æ is the area between the hyperbolic curve and the Strecke AC ! That is measured in some metric.

This projection   was dividing  the Strecke projected onto . Thus division was introduced into the arithmetic of Strecken in the form of trig and hyperbolic tables.

Few realise that Hermann introduced also the division or the product of 2 Strecken. This he called the inner product.

The Grassmann inner product is constructed by 2 vertical projections . One Strecke is projected onto the other and vice versa. The 2 divided segments of each Strecke are Strecken.,using only those projected Strecken which share a common origin a parallelogram is constructed WITHIN the larger parallelogram constructed from the initial Strecken.

The initial Strecken construct the exterior product, the projected Strecken construct the interior product.

The 2 products are related thusly

AB x BC  is the exterior product
AB x BC x cos ^2(ß) is the interior product.

They are in the ratio to the square of the trig ratios!

[jehovajah]

The significance of the last post perhaps cannot be overstated. However, if you have been following Normans redaction of mathematics you will recognise the square of the trig ratios in his definition of spread and quadrance.  What you may not know until you watch his universal hyperbolic Geometry videos is the quadrance and spread also define the hyperbolic trig ratios.

For this reason I recommend Normans work as the most fundamental implementation of the Grassmann method and analysis.

In none of these posts was it my intention to say Grassmann or Norman have got it completely correct, rather it is to point out that both of them enter into this research and work to set mathematics on the soundest footings they can construct, and they constantly attend to the foundations to make them secure and watertight.

Hermann never felt able to publish untill 1844 when he believed he had achieved something that was robust and secure. When that was assured his actual published aim was to call fellow researchers to his aid. The work was delicate and tricksy, and he could not do it all on his own.

What he did build  however has changed the paradigm of mathematical modelling in physics and may have influenced chemical notation.

I now propose to work on the text of the Vorrede and the Einleitung of the 1844 version, and would welcome discussion and meditations on those to be posted here, as well as any comment or question in the spirit of this enquiry or even good humoured comment. By god we need some! Lol!

[jehovajah]

I shall work from this text version
https://archive.org/details/dielinealeausde00grasgoog

[jehovajah]

Before I do I have to mention hermanns concept of a basis. Hermann, I think under his brother Roberts influence does not define a basi until 1862, when it is defined as an Rinheitung, a collection of monads or unit metrics. In 1844 the concept naturally arises from the law of 3 points. There we see the tru nature of a basis. The two Strecke in the law of 3 points are a journe broken into 2 stages. Thus these 2 Strecke can be used as analogies for any 2 stage process. In Cartesian geometry the ordered pair, or the sequenced pair or more tellingly the logs pair( ratio) are set to identify a unique point or position in the mosaic. They are said to be defined by the axes, but in fact this was an idea introduced by Wallis when he standardised the axes, and made his conic section equations take their iconic form!

Descartes simply took any 2 lines in the problem as fixed relative to each other, and defined all other points and lines relative to these 2 distinguished and fixed lines. In that regard Hermanns 2 Strecken were precisely the same system.

These 2 Strecken were the basis of the description of all other Strecken. Thus by working in ths way all evaluations depended on just 2 measurements.

The lines or Strecken were said to be linearly independent, geometrically this is obvious because the lines are in different orientations. The number of lines chosen depended really on choice and how much work you wanted to fo measuring.. The number of lines was never limited by Herrmann. He is united as saying no longer let 3 dimensions be the rule for space!

Because of real numbers I think I was given the notion that linear independence meant that no  " vector" in the basis could be expressed in terms of any other combination of the set of basis vectors. However Hermann was not that strict, and often worked with rational or integer reference frames not real. .

The basis Strecken then come from the notion of the law of 3 points. A representational third Strecke is added to hide the basis Strecken and these are treated as if they were Strecken themselves, not just representing a combination of 2 or a 2 step process.

This led to a distinction between basis vectors and generalised vectors, but hardly any one really explains the important distinction. The flowery set language actual obscures what it is talking about. Basis vectors are the metric vectors for evaluation ! Without them no representation can be pinned down to any place or value!

Representation theory also takes its lead from Grassmann!

Grassmann mentions in a later edition of the 1844 Ausdehnungslehre, that the heuristic process of constructing the lineal algebra was so instructive he would not change a word in the new edition. Instead he heavily annotated it.

What Grassmann showed was the way to use ring theoretical laws to construct an extension to arithmetic. In fact, any " object" that satisfied the laws of the AusdehnungsLehre , forming a ring group structure can be used as an arithmetic extension.

[jehovajah]

<a href="http://www.youtube.com/v/fEWj93XjON0&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/fEWj93XjON0&rel=1&fs=1&hd=1</a>