The Theory of Stretchy Thingys

[jehovajah]

Reviewing Normans excellent introduction to the wedge product a v b I am aware that the treatment I have exposited so far has been confusing in the sense that one is always drawn back to the " dark side"! The wedge product is inherently based on the associatve laws of multiplication. That is precisely the discussion and the exposition. So why make these distinctions?

The distinctions answer the question: where does multiplication come from? The answer is it comes from the geometrical set up of Strecken when they are being used to construct a figure on the page or a form in space!

While, in the exposition it looks like we are mimicking associative multiplication, and using the terminology found in associative multiplication the point being made is: actually, stop and think. Divest your mind of what you think you know and think this way for a few minutes: What If the symbols used to describe associative multiplication really come from describing Strecken?

The representational theory which underpins this meditative thought process links back to the form of the mosaics of the Pythgoreans, and the format of the Logos Analogos.

When Norman writes the primitive or basis vectors in the form of a matrix and then isolates the scalar factors as coefficients, he is creating a coefficient matrix which is called a Cayley matrix or table. But much simpler than that or rather prior to Cayley this layout was done by the Pythagoreans on their mosaic patterns!

Revealing this prior history by placing a wedge between the symbols is not only a small symbolic sacrifice it is the key to linking the seemingly disparate processes in mathematics into a consistent and congruent whole based on the combinatorial nature of all synthesis. It reveals the fundamental role of sequence and sequences, of order and arrangement, positioning and relativity, Before we get to summation! Aggregation and disaggregation, combining into bundles or groups ,splitting into a multiple form, all of these processes precede counting and summation.

The wedge is a simple symbol to remind one not to ignore these underlying processes in our synthetic productions.

Setting a v a to 0 is but one choice. The algebra and the arithmetic derived from this choice happens to match our consensus arithmetic very well.
But what algebra and arithmetic do we get when we set a v a to 2a?
The dot product , so called, sets a v a to  a^2. This is the basis of the Clifford Algrbras.

The Clifford algebras construct or synthesise a set of algebras based on a very complex product! Because Bill Clifford wanted to combine Quaternions with Grassmanns Lineal algebra he used a synthesis he found in Grassmanns critical review of the Quaternions. In that review Grassmann sets out a product that is based on his notation that formed a 1 to 1 mapping with the Quaternions. Once he had done that he remarked that Quaternions had nothing more to teach him!

I cannot say whether it was arrogance or just plain fact, but I do know that Hamilton regarded Grassmanns methods as superior to his own, in the sense that he recognised the greater generality of Grassmanns approach.

When I first, ever came across Grassmanns method I regarded it as "cheating"! It was plain that you set the rules to get out what you wanted! It was and is like a game in which one sets the rules, plays according to the rules and gets the outcomes of those rules.

I no longer regard it as cheating but I still think it is a very useful attitude to have! The rules set out the way you want the elements to behave, clarifies everything. What is interesting is what rules work for everyday life? What rules work for science? What rules work for wave theory? What rules work for particle physics? The list goes on.

This is a great intellectual freedom, but alo a theoreticians nightmare. Dirac for example set out rules for quantum Mechanics. Subsequently he obtained results which implied that negative energy was a factor in quantum mechanics. This was not only counter intuitive but deeply upsetting to his colleagues Mach, and Planck. They could not fault his logic, so they browbeat him into discounting his theoretical findings. In any case it was the practice to discount all absurd results of mathematics including the so called imaginary ones.

However, they went further. They actually scoured Grassmann's analytical method for years to find a way to overturn this result by Dirac. They found a method of collapsing vectors to 0 which meant that negative vectors could be annihilated. Then they rewrote Dirac's findings in these terms and got rid of the negative energy. To work, they had to accept creation from 0. This later became zero point energy, and energy from the vacuum etc..

However, Dirac was proved correct, experimentally! There did indeed appear to be behaviours that matched his negative energy. Dirac, at the time preferred to remain in the club rather than walk away from these shenanigans and he lived to see these men impeached by empirical data.

The point of the story is: you can Create any Algebra you want, but the litmus test is does it work with empirical data?

The Clifford Algebras do not always have a physical interpretation but some do. The important thing is to lay out the rules clearly so that confusion does not lead to mistakes. I find that Clifford algebras are presented in a confused way because no one really understands Hermanns work or method.

[jehovajah]

How I build a Spaciometry is from the mutual spheres that share a displacement from their centres. The spheres intersect in curves called circles bounding curved regions I call Shunyasutras. The distinguished points on these Shunyasutras fom the dual points of the circles . Taking one arbitrary dual point I consider another sphere that is now mutual to two spheres . The intersections form smaller circular Shunyasutras

The displacement of the instrument can now vary the Displacement instruments to pick out all those dual points that form concentric circles relative to the initial arbitrary points that form a circular plane. Using this circular plane I can use a pair of compass instrument to identify all those dual points that lie on a straight line called the diameter of the circular plane.

So now I can start my planar geometry with the right triangle in a semi circle!

<a href="http://www.youtube.com/v/eMt09NN1WOE&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/eMt09NN1WOE&rel=1&fs=1&hd=1</a>

Using Norman's definitions for planar forms  we now set the metric rules for our geometry. To do so se have to identify stretchy things or Strecken. Although a line has been defined , mentally we apprehend by stretches or extensions. This is how we measure!

Now the law of 3 points is applied to any 3 points on the circle . AC ~ AB + BC is representation theory for the sum of the 2 Strecken.

We will use this representation to define the general Strecken, making AB , BC  are basis Strecken .

Consequently we have to define AB x AB , BC x BC and AB x BC + BC x AB
Using representation theory we can set

AC ^2 ~c^2
AB ^2 ~ a^2
BC ^2~ b^2
AB x BC + BC x AB ~ 2 a.b cos (C)

[jehovajah]

This blogpost by Wolfram builds on the notions of the Grassmanns.
http://blog.wolfram.com/2013/11/13/something-very-big-is-coming-our-most-important-technology-project-yet/

[jehovajah]

The above representation map is not usually set out because it is not understood. I am very sure Hermann had it in the back of his mind as he progressed his analytical and synthetic modelling..

The model is a 2 dimensional map. It maps 2 dimensional objects into a 2 dimensional metric space or a trigonometric evaluation
 It works because it mas triangles into right triangles. It breaks down any triangle into right triangles and maps them to their appropriate trig evaluation.

The problem was that Hermann redefined= to mean this combinatorial mapping or identification. In his foreword and Einleitung he tries to explain these kinds of " identies" or correspondences. The word Gleichen attempts to carry this meaning. When I get to the passages I will attempt a better exposition. The German for me is not so easy but the intention shines through like a supernova!

The actual mapping is quadratic,

AC ^2~ (AB + BC)^2 —>c^2= a^2+b^2+2abcos(C)

This says that the general Strecken squared is to be thought of in terms of the general triangle formula extending Pythagoras rule. The small letters relate to the measurements of the Strecken what we now misleadingly call the magnitude, and the C relates to the measurement of the angle opposite AC which is in fact the point B!, that is the angle at that point between the Strecken AB  opening clockwise to BC .

There is an alternative form of this map which uses the Senkrecht to the Strecken AB stretched or cut which uses additional lineal measurements involved in the trig ratio for cos(C). This version highlights all the right angled triangles involved in the representational mapping.

From this mapping we can define the 3 exterior algebras and the interior products.

[jehovajah]

The first case is the primitive AB^2 that is AB x AB

This is a process of construction of a parallelogram by definition. The process is by projection parallel to the two Strecken .,however this is a degenerate case . The 2 projections only produce an extended lstreckn in the same orientation as AB. The construction produces or extends AB to 2 AB

By mapping that should represent a measure of a^2

However, it could be sensible to define this product as the 0 parallelogram. In that case we should define the metric as 0! That means we could define a^2 as 0

This makes little metric sense, but some group theoretical sense. It means that AC ^2 is 0 by definition and that AB x BC + BC x AB has to represent 2 abcos (Pi/2) to retain any sense. . But then if the Strecken are orthogonal, there is no requirement for the squares to be 0 as the 2abcos ( pi/2) eliminates the anti commuting terms and reveals Pythagoras theorem!

Instead of clarifying this twist in the rope, Bill Clifgord decided to create or intercalated the dot product from the quaternion theory, and leave the Grassmann product as apparently only defining a v a as 0!

However Grassmann was more subtle than that. Firstly because it was an extension of the associative laws of arithmetic Grassmann was able to see the representation as a Functional relationship.

By defining primitive units he was able to see the following
AC = ce3.  AC ^2= c^2e3e3
AB= ae2.  AB ^2 = a^2e2e2
BC = be1. BC ^2 = b^2e1e1

He could now define the units as forming what he called a bivector unit and these squares are base on the 0 bivector unit! The bivector unit is a separate notion to a production of Strecken. While the same constructional issue remains in that the unit Strecken construct a line 2 units long, the line is not a bivector by definition.

The product that picks out this 2 unit line can now be distinguished as a special product defined when Strecken are parallel only. This product for Strecken at an angle to each other is different to the product which is a bivector.

This special product is often called the inner product, but this is not the Grassmann inner product. It is the Bill Clifgord inner product. By these means he kept 2 distinct magnitudes in combination, a Strecken evaluation and a bivector. This was how he modelled Quaternions using Grassmanns method.

Grassmann defined it differently. He used a ratio of his outer product and his inner product to distinguish the cases! When the Strecken are parallel the ratio gives a trigonometric value, not a Strecken. The ratio can be set to go from 0 to 1 or from 1 to infinity, if allowed.

Now he arranged his algebras according to the anti commutator.

The valuation For the anti commutator  looks like this

AB x BC = abe2e1
BC x AB = abe1e2
Thus
ab(e2e1 + e1e2) ~ 2abcos(C)
When C is pi/2 then the 2 bivectors can be classed as negatives of each other and
e2e1= –e1e2
Because this sum is equated to a 0 bivector by the distributive law of multiplication. Which underpins bivector summation.
When C is 0 or pi these two sum to represent 2 or –2



There is one other question:what is the outer product? I have assumed it is the initiating vectors, but I am not sure. The outer product could be the parallelogram formed by using the vertical projectors, the Senkrecht construction or projection Strecken. It was mentioned by Grassmann in reference to his exploration of the hyperbolic functions. In addition, Norman Wildberger makes great use of it in his definition of spread.

There are still things to puzzle out, which is why I want to meditate on the text in a collaborative way.

[jehovajah]

The thought occurred:
Clifford algebras define vector multiplication in terms of
abcos ø + absinø the sum of the inner or dot product and the outer or exterior product. This may be an invention of Bill Clifford?

The contravariant and covariant tensors are defined in an analogous fashion, and this is definitely due to Ricci and Levi drawing on Peano's work using Grassmanns method .

I suspect that Hermann decided to use bivectors  bi Strecken to define the Strecken product
 Suppose A = ae1 and B = be2
Then AB is a bivector and it is defined as
  {abcos^2(ø) + absin^2(ø)}e1e2
Where e1e2 cos^2(ø) is the inner product ( Grassmann) of the Strecken e1 and e2 and the e1e2sim^2(ø) is the outer product and they are bivectors. That is bi Strecken.

The general Strecke a1e1 +a2e2 can be multiplied by b1e1 +b2e2

And factored into these inner and outer product forms
a1e1 b1e1+a2e2b1e1 + a1e1b2e2 +a2e2b2e2
a1b1e1e1+a2b1e2e1 + a1b2e1e2 +a2b2e2e2
(a1b1e1e1 +a2b2e2e2 )(sin^2(0)+ cos^2(0))+a2b1e2e1(sin^2(@)+ cos^2(@)) + a1b2e1e2(sin^2(-@)+ cos^2(-@))

(a1b1e1e1 +a2b2e2e2 )( cos^2(0))+a2b1e2e1(sin^2(@)) + a1b2e1e2(sin^2(-@)) +a2b1e2e1(cos^2(@)) + a1b2e1e2(cos^2(-@))

(a1b1e1e1 +a2b2e2e2 )( cos^2(0))+a2b1e2e1(sin^2(@)) – a1b2e2e1(sin^2(@)) +a2b1e2e1(cos^2(@)) – a1b2e2e1(cos^2(@))

(a1b1e1e1 +a2b2e2e2 )( cos^2(0))+(a2b1 – a1b2)e2e1(sin^2(@)) +(a2b1– a1b2)e2e1(cos^2(@))

Although the angle measurement direction does not affect the square of the trig functions it helps to explain why switching the unit Strecken is viewed as a negative of the initial Strecken position. It is a purely formal process rule.

The result is 2 types of inner product where  the  vectors are parallel and two bivector products one is an inner projection bivector product , the other an exterior projection bi vector product.
How we evaluate this is now up to our needs, but the coefficients, as Norman points out are "area count " coefficients and that is a useful result for all sorts of reasons, and that is why the anti commutator is retained.

[jehovajah]

The more I study these Strecken and bistrecken forms the more I see the ambiguity as flexibility . The law of 2 Strecken allows one to develop a representation theory with simple aggregation . The law of 4 points  including a third Strecken allows construction of forms and trigonometric multiplication. This allows an extensive counting of space if the bi Strecken are defined as producing parallelograms, or a calculation of areas of inner parallelograms.

The forms can be used ismply to layout the calculations inherent in the scalars or coefficients, in which case the Strecken and the bivectors do not have to have a geometric interpretation. . In certain cases bivectors degrnarate into straight line of a combined length of half a perimeter of a parallelogram. That may be geometrically useful but may not beed to be used.

Bi projection defines the inner and outer products and these can be used as a variety of given Strecken and used to form a reference frame connected to the initial Strecken.

It just seems to settle my intuition when I think that Hermann expected it to be a network of related building blocks which have to be chosen and put together with rules defined by the synthesiser.

You cannot think that Mathematics has some god given rules with Grassmann. Every mathematics is co structed by individuals or groups defining the rules of the game! That these rules can constrain a system that matches the complexity of existence is the intriguing part.

[jehovajah]

I now have to distinguish between the inner projection nd the outer projection.

Take a Strecke F the inner projection is a Strecke Fcos ø and the outer projection is a Strecke  F sin ø

These 2 Strecke are at right angles rotation to each other(pi/2).
They form a rectangular basis.
Call the inner projection a and the outer projection b
F~ a + b
ab is a bi Strecken
F^2 ~ a^2 + b^2 = (a + b)(a +b)
F is therefore a fundamental metric of the reference frame and the evaluation Strecke.
For example we could use a unit velocity vector to create a reference frame that allows changing velocity
And consequently develop a vector surface that we can use to construct
A manifold with a flip axis where the original line was placed.
The Inner Product is the bi Strecken abcos^2(pi/2)
The Outer Product is the bi Strecken absin^2(pi/2)

There are many trig relations that may be applied in this context of Strecken and bi Strecken, especially when the inner projection and the outer projection are not onto right angle axes.
http://www.sosmath.com/trig/Trig5/trig5/trig5.html
Amongst these you may recognise the anti commutator but in the context of the argument of a sin or cosine function .
http://en.wikipedia.org/wiki/List_of_trigonometric_identities

[jehovajah]

This video demonstrates the kinds of things Hermann explored in his mind regarding Strecken and bi Strecken
<a href="http://www.youtube.com/v/rz8A5l&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/rz8A5l&rel=1&fs=1&hd=1</a>
<a href="http://www.youtube.com/v/rz8A5l_yn34&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/rz8A5l_yn34&rel=1&fs=1&hd=1</a>

This is only one of the rich number of things Strecken have been used for.

[jehovajah]

This video is a solid introduction to some of Grassmann's thinking. Norman comes closer than anyone, but the commutativity of summation is where Gibbs parted from Grassmann!
The discussion above about the wedge product comes out of this distinction.
 In addition, studying Grassmann's figures reveals even greater differences in thought!
Unfortunately we do not see a full set of figures, so if anyone can post a copy I would be grateful!

<a href="http://www.youtube.com/v/132amJvoLpU&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/132amJvoLpU&rel=1&fs=1&hd=1</a>
<a href="http://www.youtube.com/v/132amJvoLpU&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/132amJvoLpU&rel=1&fs=1&hd=1</a>

[jehovajah]

I forgot about this lecture by Norman. Much much more careful presentation .

<a href="http://www.youtube.com/v/TuaInpbbsg4&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/TuaInpbbsg4&rel=1&fs=1&hd=1</a>
<a href="http://www.youtube.com/v/TuaInpbbsg4&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/TuaInpbbsg4&rel=1&fs=1&hd=1</a>

[jehovajah]

I do not recommend the following, but link them so you can see how some mathematicians complexify what is essentially a very simple notion,
<a href="http://www.youtube.com/v/jQ&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/jQ&rel=1&fs=1&hd=1</a>
<a href="http://www.youtube.com/v/K5d6FuQYyEU&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/K5d6FuQYyEU&rel=1&fs=1&hd=1</a>

I need to see copies of Grassmanns figures at the back of Ausdehnungslehre 1844, please, anyone.
What I have seen leads me to a nice concept of the circle and unit sphere trigonometric ratios underpinning Grassmann's initial notions., but I need to see the figures to meditate further on this point.

In the meantime I hope to start work translating very soon, and would appreciate all feedback on the translation.

It won't be an ordinary translation but a meditative one. Anyone who speaks, writes or thinks German can contribute!

https://archive.org/details/dielinealeausde00grasgoog

[jehovajah]

AUSDEHNUNGSLEHRE 1844

The Expertise of the extensive Magnitudes
Or the theory of Extending Magnitude
A new mathematical discipline
Set out according to the discipline  and through applications exposited
By Hemann Grassmann
Teacher at the Friedrich-Wilhelms-School in Stettin(Szczecin)
First part containing
The Lineal extending magnitude Doctrine

The Theory of Lineal extending Magnitude
A new offshoot of mathematics
Set out according to and through applications on the other remaining branches of Mathematics.
How also
Statics,Mechanics, the Theory of Magnetism, and the laws of Crystals are exposited
By Hermann Grassmann
Teacher at the Friedrich-Wilhelms-School in Stettin
With 1 table
Printer Otto  Wigand
 Leipzig.

[jehovajah]

It seems some pages are missing so I recommend the following later reprint with annotations

http://www.uni-potsdam.de/u/philosophie/grassmann/Werke/Hermann/Ausdehnungslehre_1878.pdf






Some comments on applications to physics
http://my.ilstu.edu/~lmiones/Math-Phys-Seminar/M4PS2-EC-Introd.pdf

[jehovajah]

"Grassmann, Hermann Günther." Complete Dictionary of Scientific Biography. 2008. Encyclopedia.com. (December 18, 2013). http://www.encyclopedia.com/doc/1G2-2830904887.html

Without annoying advert
http://www.encyclopedia.com/topic/Hermann_Gunther_Grassmann.aspx