The Theory of Stretchy Thingys

[jehovajah]

Commentary

I actually came to note that the term "geometrical product" as used by Grassmann has a foot noted explanation. It refers to trigonometry and space theory.

http://my.opera.com/jehovajah/blog/2012/02/25/the-complex-numbers-arise-out-of-3-mistakes

This link is not one of my clearest blog posts, but reflects the meditative digestion of a lot of new information at the time. The basic thesis was all of mathematics is based on trigonometry nd trigonometric ratios!

While I have not read the referred to text, I skimmed through a German text of that era dealing with space theory in the style of Grassmann! I can not say which Grassmann it was alluding to, but I guess it was Hemnn.

Space theory ( Raumelehre) seems to be equivalent to the concept of topological or metric spaces. The essential idea of a metric space, cutting through the jargon is : you and or me can measure our relative positions! Well, when you look at it like that one can realise that trigonometry , classical spherical trigonometry was devised by Hipparchus I think to do precisely that!

Measure theory, Lesbesgues measures etc deal exactly with this same issue that in a real sense spherical trigonometry solved millennia ago!

So the term geometrical product, in Grassmanns mind was a reference to the products common in trigonometry!

Most of us learn the basic rules for calculating areas without ever being told these are geometric products. If you get on with trigonometry and some do not, you may realise that these products have a deep trigonometric relation, due to the right triangle. Again you would not have been told, these are geometric products! I just call them trigonometry.

So it is easy to see why the notions of geometric products seem so alien and yet give such familiar results!

Actually, as I have cleared my mind from imposed distinctions, a process Grassmann deeply refers to in this text, I realise more snd more that this text is about basic arithmetic and fundamental trigonometry.

Why does it appear so revolutionary? And why does Hermann give it such a big build up?

Well my guess is that when you cut through all the BS jargon and find it is really a few principles used iteratively , that is over and over agin at different levels, it is a big deal!

For me at this stage, just thinking about a line segment as the basis for all this analytical superstructure( and I have not even got to chapter 1 yet!) is liberating. Die Begränzte Linie! Die Strecken. It creases me up when I think : is that all it is? Lol!

Aside from the deep symbolism of the line segment, an ancient Greek Notion, Pythaoreans certainly, the inevitability of a line in my mental analysis of space is unavoidabl. I do wonder how a blind person conceptualises space, and in that regard, symbolic algebras take on a new and practical sense.

Braille and musical dynamics provide an alternative symbolism I imagine for what must be a different proprioceptive experience of space.

Oh by the way, memory and consciousness is another fundamental duality like line and point or region and space.

Simply put : Grassmanns method of analysis and synthesis applies trigonometric products and sums to the space in and of which we have our very being. As such it is a powerful but limited description of our experience unless fractally iterated!

[youhn]

This is the original reference I stumbled on.

http://www.argyrou.eclipse.co.uk/GreekEtymology.htm

And this was the blog post it inspired!

http://my.opera.com/jehovajah/blog/2012/03/04/algebra-always-has-hurt.

Thanks for the references! I do understand the more popular things (like meaning of words) are mostly shallow and not always true or complete. This "pain of mind" reminds me now of some explanation of Arthur Benjamin ( see <a href="https://www.youtube.com/v/M4vqr3_ROIk&rel=1&fs=1&hd=1" target="_blank">https://www.youtube.com/v/M4vqr3_ROIk&rel=1&fs=1&hd=1</a> from around 12:30). My math is above average (lol, just image average), but I still count myself at the 95% you write about.

Well my guess is that when you cut through all the BS jargon and find it is really a few principles used iteratively , that is over and over agin at different levels, it is a big deal!

This really sounds fractal ...

I could go on, but I will stop here!

Thanks for that aswell! 
You can be very long in your writings.

When math and programming overlap, I do find myself struggeling between using (greek) symbols and semi-full names. The first keeps the code "clean" and easy (faster) to write down. The second results in far more text and symbols, but do keep it on a more understandable level. Seems you prefer to stay away from much symbols ...

[jehovajah]

Ausdehnungslehre
The remainder was revealing itself in due course, that with the appropriate specifications how they themselves will be found  in the work , the intersection point of two lines, the intersection line of two planes and the intersection point of three planes could be apprehended as products of these lines or of these planes, then next  identically a highly simple and general theory of curves, was revealing itself from this.

Thereupon I now went over to the wider application  and  the fundamental ground of it, which I have in mind for a second volume, where I specifically have critiqued all those  things as a group  , which fundamental ground presupposes all similar ways to the handles or labels of pivoting( swivelling round) or angle,
 There in the second volume , which will complete the work , it should appear later in print for the first time, so the need  seems to me for the general view of the Whole work, which is to here signpost to some exact form the appropriate revelations.

[jehovajah]

Youhn,  "windbag " could be my middle name! Lol!

In my defense, I have to say I have an autistic trait, one of several, called polylogia. I either say too little and so become obscure, or too much obscuring what I am saying!

A lot of my writings are effectively machine dumps!

I hope some find something useful in them, but if not I do not or cannot care, because I have to get them out of my processing system to move on!

With regard to your programming style: I opt for meaning over symbol. Machine processing today is so powerful it hardly makes a difference. But for a human legibility is crucial . Commenting is an alternative strategy, but I find the use of symbols with commenting is a compromise with little machine benefit.

Because I prefer this style I write shorter function or procedure calls and call them in main almost like a translation. Thus, in c or c++ I used to write a main that looked like an explanation of what it was doing!

I am not up to date with current programming practice, but I did like the object oriented concept, and I loved pointers. These gave me greater freedom of expression than function calls.

I never did any real machine code programming, because I was not a technician, but I appreciated just how hardware dependent it was when I devised my own pointer calls for Qbasic, using peek, poke and LRecord.. It was fascinating, but I almost lost my wife while I was lost in that deep world!

[jehovajah]

Arthur Benjamin very entertaining! Lol!
In the Indian subcontinent Vedic mathematicians demonstrate such skills, and certsinly our mathematical greats had this ability: Newton, Wallis Cotes, Kepler,Euler, to name but a few.

While it was mysterious in the past it should no longer be mysterious to us. This is a neurological processing trait that some of us have. We probably all have some ability to do this but not as efficiently as others.

The neurological programming starts with counting using fingers or digits usually. Most of thus are then bullied into dropping this neurological association and forced to use something usually called, but never explained , " rote learning! "

This rote learning is actually quite brutal! It consists of continually disciplining the processing centres until they respond in only one way.. For about 90% of us this is where we learn to hate mathematics ! On the other hand most of us love counting and singing and dancing in one form or another! Neurological processing is so much fun!

Of course we won't all be able to calculate at that speed, but we would all be better at calculating and all happier!

[jehovajah]

Commentary

Grassmann here lays out the summary of the current work and then describes the plan for the second volume!
The work here he saw as revelatory and progressive in the way he felt it clarified and simplified much of geometry and Mechanics. His second volume would be more stringent , formal and exact. However the underlying principle to him was the dynamic rotation and the angle!

This is a familiar analysis to me. One could rephrase this as " it's all about trigonometry! ".

I came to the same conclusion by the end of my research in the Fractal Foundations thread. I appreciated, but only to some extent, how vectors, so called were a form of trigonometric summation involving the constant i, but I had very little clarity on that constant at the time. I have a settled opinion about it now, after researching Bombelli, Hamilton and the Greek forefathers.

I know thst Grassmann read widely and deeply including Lagrange. Lagrange adopted most of Newton's approaches, including the use of i, as did Cotes through De Moivre. As Mechanics the i had the significance of an arc measure. Cotes said as much in some work he was doing on integrals. Unfortunately he died before he could write his Harmonium Mensurarsm, which I think would have revealed the Concepts Euler put forward some  decades later.

In this case Grassmann is alluding to the dynamic and rotational nature of the Barycentric calculus, as a foundation to all the results he was revealing. None of his Strecken were conceived as stationary. They always pivoted about some fulcrum! Thus he could transform from one coordinate system to another, simplifying any description by an appropriate choice of reference frame.

The use of the product rules in addition meant he could work with points, convert collections of points into lines, planes or volumes, product these to get planes , lines or points, and so move surely through a complex space description symbolically.

The theory of curves he refers to has piqued my interest. I wonder if it is a version of the Bezier et al curves?

[jehovajah]

" which fundmental ground presupposes all the similar ways to the handles for pivoting or angle".

I tried not to presume too much, but was constrained initially by the concept of vector, and its issues. Meanwhile I kept reading of a great freedom in text . At first I thought it was to be found in the rectilinear foms and constructions. I could not understand how Clifford Algebra brought trig ratios into the labels.
Then I found the inner product relied on the cosine tables squared to evaluate.

Eventually I started to explore the trig ratios in relation to Grassmanns statents.

This sentence explains all, to me. Trigonometry is assumed because it is in fact fundsmental to the whole method. On top of that trig was deliberately left out of these first presentations to make the revelation more dramatic! The exact formulary would be presented unequivocally in the second volume..

Clifford therefore brought the trig ratios out of the cupboard ad made them plain, but lost the "purity" of line and point product.. Instead he replaced it with an actual product of lengths and trig ratios.  Then the dot product, the outer product, the combined product all tend to be confusingly presented. Terminology is changed and the concept of ponit ,line and plane product seems to get overtaken by some other concepts.

Now I know that the trig ratios are key everything starts to fall into place, especially the intimate link to Laz Plaths Trochoids for the Dynmical pivoting Grassmnn clearly had in mind.

The work is clearly focused on cells or regions that tessellate. The work of Escher show how iterated small changes can result in detailed descriptions .
http://en.wikipedia.org/wiki/File:Escher,_Metamorphosis_III.jpg

[jehovajah]

Commentary
Now, when I was a lad!

In primary school I was taught to draw a line horizontally and mark 2 points at a given displacement from each other, the beginning and end of a length on a ruler, even to name them A and B. Then I was taught to use a semi circular protractor to mark a third point , which was a construction point. This point was approximately at the radius of the semi circle, but was never measured. The aim here was to construct an angle, so a ruler was used to try and draw a line through A and this construction point.

Well, we were told to use a sharpened pencil, to look very carefully in a certain position to the ruler , to hold the ruler still, checking all the while that the construction line went through the 2 points. This was definitely a skill honed by practice.

Length and angle were the first geometric constructions taught and eventually mastered. In the process I absorbed a reference frame for the page called horizontal and vertical.. Angle was used rather than direction or orientation.

The next skill was the compass. This was a trickier skill for drawing arcs and circles. The paper had to be kept still, the compass rigid, the point stationary and not ripping the paper and the pencil fixed in its screwed holder and sharp!

Very tricky!

Later I did some technical drawing and was introduced to the costly equipment designed to minimise all these issues, but by then I already had fixed ideas about Geometry. Using a compass was somehow better and more accurate than measuring with just a ruler , even though you had to use a ruler to set the displacement in the first place!

Then I learned the parts of various figures , and how to construct them with a ruler and compass on the page. No 3d constructions were ever done! I did woodwork instead and used carpenters tools to construct face edges and planed faces.

All in all a neurological understanding of constructing forms in space, based on a hodge podge of geometrical ideas and theorems, especially Pythagoras. Everything was applied but never connected by any theory I knew. It was just a collection of skills and methods I learned to call upon to construct forms.

In primary school I learned by rote that area was a calculated thing. I had no idea what area was, just how to calculate it. Come to think of it I had no idea what length was or what angle measure was. I had tools . These tools defined length, angle and later area when I was shown how graph paper divided surface into little squares!

Graph paper, ruler and protractor were my fundamental tools with a pair of compasses to draw arcs. Arcs could not be accurately measured I was told, that was why we needed to calculate pi!

Formulae were introduced. Sum of angles, perimeters of forms , another sum. And areas of forms, these were called products.
So we had sums and products for these geometrical forms. I just had to learn area products which initially were just base times height, 2 lengths producted as long as they were perpendicular. Perpendicularity was the concept of relative orientation , direction still did not enter the geometrical world I had developed until we started to do navigation problems!
Base times height covered the are of all 4 sided figures with parallel lines. 1/2 base times height covered all triangles.
Complex shapes were solved for area by a sum of products! Usually two lengths producted and summed with another 2 lengths producted . The complex form was broken into forms accessible to the formulae , usually triangles and parallelograms!

This was not connected in any consistent theoretical structure to my knowledge.

Curiosities were brought in such as Herons formula, and just learned but never really understood . Finally the trigonometric rules were learned and the relationship between the circular geometry and the internal cyclic quadrilaterals and triangles . This was presented as an overall set of rules, but no general theoretical structure was alluded to with the proofs..

From this survey I hope you can see that the components of Hermanns Ausdehnungslehre, the Ausdehnungsgroesse are common in Trigonometry, even the sum of products. However, here the Barycentric calculus and the Grassmann analytical method advance the subject  by providing a unifying framework to the methodology!

In so doing Grassmann introduces a " fancy" or a Förderung, that he promotes. This idea is that the product and sum concepts extend to all the different types of magnitude, namely point, line ,plane, volume. We really do not have n dimensional magnitude concepts or experiences, and yet this powerful conception organises our thinking even in that " inexperienced, imaginary " realm.

To show his idea and to make it work he had to focus on how labels were and are defined.

The second thing he did was quite obvious but overlooked: he kept all information on the page in symbolic form, and specifically as a product form he kept length and associated direction  ( orientation) tigether(Festhalten).

Using this he picks out the fundamental group and ring structures required to make this work as an algebra. In so doing he generates not one but many algebras!

Now he has a consistent method of Analysis he can use that to synthesise or construct these algebras in detail. Each detail is different, but the whole benefits from a consistent notational concept and the group and ring theoretic structures.

Now in mathematics we can pretty much agree that it is about aggregating things, and the principle aggregation methods are summation and production! Hat we do not always realise is what production is, and where it comes from.

Production comes from division or disaggregation. This seems counter intuitive, and it is. Most philosophical mathematicians would opt for production as repeated aggregation. However, nature shows that production is based on division. Once division has occurred aggregation follows. This aggregatin is regionalised, which is why division is crucial as a first step to production!

In book 7 of the Stoikeia Euclid lays this out as best as he can unfortunately translators have confused multiple and multiple form with multiplication. Hamilton in his paper the theory of couples or the science of Pure time makes this distinction clear, but it is lost on the reader who believes multiplication is as taught through multiplication tables! These really should be called factorisation tables.

Thus Grassmann corrects this fault in the process of developing his analytical methods and the product is freed to be a multiple form, once again!

For example, finding the area of a rectangle is finding the multiple form that covers the rectangle precisely. The form can be little squares or little rectangles or little triangles. The area is the count of these little forms.

Hang on a minute! How do we get the area of these little forms?!!

That dear reader, is the reason why all trigonometry is fractal! In the past we stated that we defined these area to be Monas or unit, that is 1. Today we look past this statement without realising that ultimately our models are not based on any ultimate standard, but on an agreed convention of a standard! If god or nature does not agree with us, it is up to us to change our definitions accordingly!

Euclid and the Greeks knew this, and so they framed their philosophy so that it applied to whatever standards or conventions we could ever make. Today this is called " gauge invariance" or even gauge symmetry. Conservation laws particularly are looked for wherever gauge symmetry can be identified. Hermann called this " Keine Abweichung". More importantly, this was invariance achieved in a dynamic rotational and translational setting!

[jehovajah]

Commentary

Now the initial whimsical fancy Grassmnn started off with .
AB = –BA

This makes geometrical and trigonometric sense, but no product sense as yet.

Then  AB + BC = AC .

This makes trigonometric sense in only 2 cases. It makes no geometric sense outside these 2 cases. Why define it as a general rule?

There is evidence in the figures that Grassmann considered this from the poit of view of relative rotation about the common point. In that case the statement is trigonometric ally true if the lines are free to rotte into a straight line. Thus he might have from the outset considered these commonly joined Strecken as rotating hands of a clock evaluated in a straight line . His notation at this stage wold then imply rotation, with a stretchy elastic bnd AC joining the ends .

This is a fanciful notion indeed, and of little trigonometric value beyond implying that Pythagoras theorem would need to add or subtract a term to make this true.

In the second stage, while looking at trigonometric products he realised that in the general quadrilateral, to find the area one needs to divide the shape ino triangles. This is a procedural step, purely intuitive and not trigonometric in any way. But if one looks at the rule it suggests that 2 joined sides should be replaced by the third line segment joining their end points!

Thus the fancy suggests a technique of aslysis: divide complex shapes into triangles by using the appropriate directed line!

Again in the trigonometric product for area, two directed lines in a fan shape suggest a product for a parallelogram .

Thus the notation is gradually becoming suggestive or representational of how to proceed with an analysis!

Then when he saw the trigonometric notation where the points of a form are joined by lines to which a lowercase letter is assigned to represent displacement measures, it suggested that the 2 points ths elves were producing to give a line segment!
Finally ombining the measure with the point product gave him a symbolic representation of ny line segment.

But further support for his himsical notion, the idea he promoted comes from the use of coordinate systems. This was a later realisation that modified his thinking. Using the orthogonal axes coordinate system he could represent the three points by projections onto the two axes in the plane.
Thus every point was a vertex of a rectangle. But for every point the measurements were projected into straight lines. In that case his equation was true in those straight lines! Thus the combination of those projections wold also be true, that is they would validate that the displacement between the points would sum like the general rule. The actual trigonometric length would use Pythagoras to calculate it using the information stored in the points, or point representation!

This whimsical idea turned out to be the way to record the subjective processes we humans use to analyse and synthesise a spatil description!

In that regard it is a remarkable demonstration of belief in an intuition, against all odds! Grassmann had notated a simple way to include or engage our human software in the best practice for analysing spatial descriptions. He had found a heuristic methodology that was consistent and consistently optimal! Moreover, he persued it with fortitude and conviction to demonstrate his claim that it was an expertise that would give rise to,new branches of Mathematics. To prove his point, he single handely created one! The lineal Algebra.

[jehovajah]

Bearing in mind that Norman bases everything on natural numbers, remember that for Grassman the line and its length are locked together. Thus a number is actually a length of line/ line segment

Now in this case , using orthogonal coordinates as primitive lines, at right angles to each other. These then product to form rectangles or parallelapioeds. Everything else applies as Grassmann conceived it.

<a href="http://www.youtube.com/v/dVk3CpjHR4Y&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/dVk3CpjHR4Y&rel=1&fs=1&hd=1</a>
Don't believe the misrepresentation of Euclid either!

<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>
Actually that product rule is now, by Grassmanns modifications misleading! The product of 2 lines is by now a point! So what we now have is the product of 2 points being the primitive x axis displacement and the primitive y axis displacement( 2 orthogonal line segments) . Then these 2 line segments are projected by each other onto parallel lines through the end points. These lines then product to give the required point of the rectangle.

Thus in Grassmnns thinking projection is another fundamental process. There are 3 types of projection: parallel, point and circular. Both circular and parallel require 2 initial points( one line segment) while point requires 2 line segments with a common point.

The projection line will be what was traditionally called a construction line or arc.

So far straight line segments have been catered for by Grassmans point product notion. Nothing has yet been defined for the arc, yet, but it was definitely under consideration under the Schwenkung and angle notation and research. The labels for swivelling have been mentioned twice so far.

Note we cannot escape from the point product, except by 0 length! Thus in many descriptions one factor of the point products may be " known " or given, the other has to be found by a series of product and summation manipultions.

By this Geometry is reduced to a computational framework.

[jehovajah]

Heap sorts, binary tree( binomial structure)

<a href="http://www.youtube.com/v/GreOd9DzZn0&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/GreOd9DzZn0&rel=1&fs=1&hd=1</a>

[jehovajah]

There is a deep structural connection between the binomial expansion and all of computational mathematics. Similarly Grassmanns scheme would relate to a deep connection between the trinomial expansion and his insights.

Few, perhaps realise the trinomial can be modelled by a 3 dimensional or tetrahedral structure. The symmetries therefore can be quite profound.

Normans treatment of polynumbers uses this heap structure as a polynomial structure for the concept of an arbitrary magnitude. Consequently the terms variables, unknowns  can be rep,aced by a data structure based on the binomial heap. Note the polynomial structure of the levels. .

The important point is the binomial expansion encodes or includes the choices of combinations for the product factors somewhere in its structure.

[jehovajah]

Commentary
Grassmann advanced his ideas and his Förderung. He was not creating lineal algebra, he was creating a sound method of analysis and synthesis.

As a student learning mathematics he developed, as we do, his own method of understnding nd memorising the material. We all have our own idiosyncratic memory techniques, mnemonics, states of mind, lucky items etc. for rote learning we all have our little lins and hooks that get us to the next line or through the next stanza. We are actively engaged in how we learn and how we remember.

Unfortunately this is all schooled out of us. We are forced to conform to some standard that reflects the teachers understanding, not ours! We learn pretty quick through praise and rebuke what the teacher expects. Unfortunately what the teacher expects and gets more often thn not, is not necessarily consistent with our experience. This is where confusion sets in.

How can there be more than one right answer? Well there usually is, but some teachers do not let you know that. It is worse in mathematics! Some pedagogues actually boast that they like mathematics teaching because an answer is either right or wrong! They present a digital view of mathematical correctness.

Mst of us in our bones know this is not " true"! However we learn to dread this draconian power the Maths teacher has over our developing understnding of counting and measurement. For most of us this purposely grnerated nicety is too much! We learn to hate the subject.

What if some child came up with an alternative way of counting, using different names rhythms and images? Would a Maths teacher celebrate? I suspect not! But the music teacher probably would, the dance teacher might admire the creativity, maybe the cultural aficionados might hail it as a major breakthrough!  All these possibilities are so easily crushed with a cross by some myopic Maths teacher.

Thus it require intestinal fortitude on the part of a child to believe in nd promote its own vision of how to o things over the consensus. Much hardship has to be endured, much misrepresentation and misunderstanding.

Hermann saw things in a slightly if fervent way. He did not alter trigonometry or space measurement, he simply relabelled it. He relabelled trigonometry in such a way hat he did not need to write down reams of notation or keep looking up to orient his description according to a diagram. All the information was recorded in the symbols he set down on the page.

By inspecting the symbols he had all the information he could ever need and more!  By manipulating the symbolic positions he could identify which lines he needed to focus on. Once that set up was done he could write out the tigonmetric um that gve the result!

Yes this seemed to be a quicker way to o these calculations. Hermann did not say that. He said it was less wordy and more compact! Thus he could write a proof in a few lines!

Have you ever been fooled by those ads that say build a " whatever" in just minutes? Well that is what Hermann is doing here! Write the celestial Mechnics of LaPlace on the back of a postage stamp!  Of course the meditative processing that is involved with analysing these symbols to see which should be combined could take hours days or even weeks!

The ,ore familiar you are with ny system, the quicker your expertise shows itself. Grassmanns methodology for his analysis and synthesis does not exclude the work of set up and adjustment to the new description. However, once that is done and agreed new results consistently follow!

Grassmann methods apply geometry and trigonometry in a new consistent notation which vntually leads to sound computational results related to mechanical reality.

Cn we replace  empirical experimental and observational enquiry with mathmatical Anlysis and synthesis?
No!

But we can use the 2 in tandem to build better models, that compute accurately what we have soundly described geometrically and trigonometrically.

We have to first get the mechsnics right o get the geometry right to get the correct trigonometry. Once that is done, we have a sound model we can explore theoretically as long as we like. This model is not " the Truth", however. It is jut oe model of many we could and can now build using computers! The human battle for dominance and truth takes us out into the big bad world of religion and geopolitics!

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

[jehovajah]

Ausdehnungslehre 1844CONTINUE

To this end, I have  given to you firstly the results which had  alteady revealed themselves before the co dependent processing( melded reworking) . I have flatly shown  how the product of a  pair of line segments can be considered to be the parallelogram, even if  specificlly as everywhere happens here, the direction of the line segment is  firmly fixed to it. But then how this product is distinguished through this (requirement) : the sign of the product has to switch as the factors are switched,  while at the same time it is apparent The product for a pair of collinear parallel line segments in the same direction is null.

To This labelling is related another set of labels at the side of this one, which likewise is related to the line segments with a direction firmly attached . Speciically, if I project one line segment into the other, I put an arithmetic  product in the line segment projected onto. Likewise the product of those kinds of line segments have this arithmetic product in, in this case , for this reason also the multiplicative relationship for the addition applies. However this product is now of a different type to the first, in that switching the factors does not switch the sign , and these kinds of line segments have a null product when they are directly perpendicular to one another .

I named the first kind of product the outer product, and the last kind of product of line segments resulting from onto projections the inner product,  and so for the outer line segments, because only by stepping farther apart in directions; and so  for these inner line segments only by  stepping closer to each other in directions,they had an assignable Value
 that means  
•to evaluate by ratioed amongst one another products always had a  value that could be found( in a table).
•There is always a value between the 2 products when they are put in ratio or divided.
•By partially being nested


Through my working through  the Mecanique Analytique I  already had pointed out to me  these kind of inner products as Essential! At the same time it lead to the label of the absolute length.

Footnote
Compare with the copy of chapter 3
Compare with the same chapter.

[jehovajah]

Commentary

It is instructive to look at the sums and products Grassmann has introduced by his Förderung!

Ths sum of 2 joined Strecken .
AB + BC  = AC

This sum relies on the joint point B so that the line segments can rotate relative to one another Ito a like directed or oppositely directed line segment pair. Only in these 2 cases is the sum correct for the lengths of the line segments AB, BC, AC .

However this sum is the main plank of his Förderung and he defined or set it as a rule that is always the case in a specific sense: that of relative displacement. It is a quite different rule to the trigonometric rules Sally associated ith geometry.

The first product is AB = – BA

I am not sure that Grassmann initially saw this as a product. Apart from the minus sign it is standard geometrical notation for describing a line. I am not even sure if it was used to describe direction apart from the specific act of construction when the direction of producing a line( extending it) . Certainly it is a convenient shirt hand, but as written it was meaningless until Möbius set out his rules.

In this sense it is an invention of Möbius not Grassmann , and similarly the um of 2 Strecken idea owes a lot to Möbius. However it is clear Möbius did not have this precise summation rule for Streken.

Although this is a product rule I think Grassmann states he did not realise until he had his revelation regarding the trigonometric product formulae.

The second summation rule comes later after the point product realisation and is the Schwerpunkt or Barycentric summation rule
mAB + nBC = pAC

m,n,p are lengt factors, A, B, C are point factors

In this way he combined lengt and direction into the notation for a line segment. After this, whenever he uses Strecke or line segment he means this triple product combination, with a sign switch included if the point factors change position, but not if the length factor changes position.

Initially Grassmann introduced the line segment product. This was to be considered as generating a parallelogram . Later however he introduces the point products. Now 3 points produce a flat space or flat figure.mit is called a Fläche not a parallelogram. Those same three points generate 6 Strecken of 3 contra direction pairs( binomial coefficient), thus 3 positive signe parallelograms and 3 negative signed ones,ndepending on whether the fan product AB . AC  or the wedge product AB . BC  was used to prod cue the parallelogram.

So the point products produce a triangular form in space, while the Strecken products produce parallelogram forms in space.

This is not yet clear to me but it is clearly set out regarding the products of points, so I was surprised to see him seemingly revert back to a Strecken form until I realised it was my understanding which was at fault.

For example the intersection points are not defined in terms of Strecken but in terms of lines and planes. In fact the intersection products are defined in terms of primitive concepts for which as yet Grassmann has given no product definition! There is no product definition of a plane or a line or a point. These are primitives on which he build some product definitions.

Now to this collection of products he adds the inner and outer product for 2 Strecken by this stage it is important to note that a Strecke while being a line segment now has a complex notational form!

So cAB = –cBA is the basic product label for a single line segment.

The outer product for a parallelogram would be, say for a fan product

cAB.bAC = cbABAC

Swapping any of the point factors around would introduce a sign switch. It is yet to be revealed what AABC produces! All I know is that it is negative to the parallelogram. Of course BAAC is a wedge product parallelogram and again negative in sign.

The inner product is different AC is vertically projected onto AB . AB might have to be produced to e to meet the foot of this projection. But that is the same as increasing the length factor of AB so AE = qAB say.
In addition the length q is an arithmetical product of the length of AC = b! The full descriptor being bAC

q is set to bcosø the angle measure between the line segments. Thus the line segments project this arithmetical product into each other. This is an important distinction in the types of line segments!

Grassmann states that the two types should be written besides each other. I am not sure if this is a recommendation to save time or part of his method.

For example cAB projected onto aBC produced give 2 more associated line segments  c cosøBC and acosøBA.

If BA has to be produced to project BC onto it then cosø will actually have to be negative. The 2 negatives convert to a positive. These types of Strecken can be negative or positive depending on the angle between them, but the product of these vectors, the inner product is always positive

The product of these vertically projected line segments is cacos²øBCBA

Looking at this product one can see how switching the Strecken factors change also the direction of the angle measure! This change causes the Strecken sign switch to be switched back.,also the projected Strecken when at right angles give a null product.

Grassmanns inner product is different to the dot product or the Euclidesn dot product,

Also the outer product is proportional to the inner Productt in the ratio 1:cos²Ø

Kram1032 found this resource:

Grassmann Algebra in Game Development (slides from a talk at GDC'14