Grassmann chose. Very direct notation for a point A a line a and a rotation v clockwise and ^ for anti clockwise.
So a = AB means a represents the line which displaces from A to B, and has the orientation of this line.
Thus I my denote the contra displacement by –a
And –a = – AB = BA
There are several immediate corollaries to that statement.
Now for a pair of lines a and b
a ^ b represents a rotation relative to a
So that the contra rotation –(a ^b) is b v dot
Thus a ^ b = –( b v a)
But it is also the case that
a ^ b = –(a v b) and so
b v a = a v b
This indicates that commutativity involves referencing 2 different routes or kinds of rotations and the Equals sign has to change its meaning or denotation.,to make this clear Grassmann referred to the exterior product and the interior product., but for this case he called it an exterior rotation and nn interior rotation . The equal sign the result is the same as, not the processes are the same. Some have called it a regressive algebrs, others recognise it as a conjugate algebra.
Having defined these product processes, the actual product is a form called a bivector by.some. Grassmann thought of it as a Flache a flat parallelogram . This diverted him from the conjugate slgebrs and lead to some theoretical difficulties. …
The commutativity example above has given me some intuitive inease. I laugh because Grassmann had the same reaction to the anti commutative or non commutative case.
Th issue is intuition. By thi I mean that collection of habits, experiences and perceptions even unconscious processes that signal when something needs a closer look!
Grassmann gave Heuristics as one of the reasons he did not change a word of his 1844 masterpiece, but I suspect that besides pride, the forced redaction of his work by his brother Robert, left several raw memories of subtleties thrown out by the concise mathematical format expected at the time.
Heuristics is however a hugely important reason. It speaks to how, why and when a problem solver comes up with a solution. It is a Greek word denoting the efforts of the heroic pedagogues of the Aristotelian school of thought, and it is associated with the etymology of words nd ideas.
So the problem is the rotation notion has more degrees of freedom, so to speak, than the lineal translation or displacement notation. The fact that I have to step outside the object and observer framework I started with, and introduce some related but grander ideas is indicative of the difficulty. Suppose I did not have this alternative framework to view the issue from, I would be intuitively nd heuristic ally stuck!
Many mistake the algebraic symbols for algebra, whereas Algebra is the heroic which the symbols act as mnemonics for. That rhetoric has to be "logical", that is systematic in the sequence of the process. The issue is that although the symbols are fixed, the combinatorial sequence of them implies a form of Rhetoric which is not fixed. I am free to expound the rhetoric in any number of ways and unfortunately some ways are inconsistent with other ways!
If a number of rhetorics are consistent, they are often said to be true thereby, or even "The Truth"! I have no other way of determining consistency accept by comparing , contrasting and concluding. These are heuristic thought patterns, and they very much are involved in the Grassmann Algebra, if not in all algebra..
The trouble with " Mathematicians" is they want to get it right! They do not want to do philosophy or consider probabilities, they just want to be right. Consequently this breed of Astrologer divorces itself from the philosophical enquiry of the ancients and attempts to construct defensible impenetrable towers and castles from symbols.
To be a Pythagorean Mathematikos you had to be able to engage in the philosophical enquiry. So in fact Grassmann algebra does. His brothers 1862 redaction does not. Which one is called " mathematical"?
Returning to the Strecke. It is defined by two end points and all the " dual" points in between, called " right or straight". Grassmann does not start here, as a young man, creating the algebra piece by piece. He takes for granted what a Strecke is, using his Father's definition. But what he notices is that the terminology is dynamic!
A is not very dynamic , until you realise it could be any point! The actual point is arbitrary. Thus B is dynamic. And AB is dynamic, it could be any Strecke in any orientation, but what was now fixed, in the Newtonian observation was a displacement direction. AB as a direction of displacement was dynamic , but it was a fixed process: you had to go from A to B.
So now we are describing a dynamic process by a Strecke. The Strecke in front of us is a symbolic mnemonic for a dynamic process. The resultant or out put of that pr ocess, the product of it was that specific line segment, as only one instance of an arbitrary Strecke.
So the contra process BA carries a – symbol to highlight it is contrary as a process to another process it is Equated to. In all the geometric or process algebras it is the = sign that is the sloppiest symbol. Mathematicians will not rid themselves of it, nor of the notion of equality.
In many rigorous treatments, identity, duality , congruence, imilarity are used in place of equals and equality. It gains it's power and influence from the false certainty of the number concept.
So the lines a and b are actually resultants from a Strecke process corresponding to each. Grassmann now wants to explore a rotation process. In this process the beginning and ends are lines, the resultant is a relative orientation, but the dynamic process is the animation of a line rotating relative to anther line. The common point of rotation is not specified in the arbitrary case, but in fct Grassmann starts with the case where the meet of the 2 lines is specified as the centre of rotation..
Immediately we have 2 kinds of rotation and one is contra to the other as a process. The process produces a resultant animation that clearly goes in contra direction. In this case the displacement is a rotation! The contra rotation naturally carries the _– symbol.
Now we come to another number issue, negative numbers! From these we learn our bad habits about the – symbol. We start to call it " sign" and we begin, mistakenly to invest it with an operator concept. This screws with the algebraic degrees of freedom, because it ties an arithmetical" operation" to sn algebraic exposition. It was Hamilton who introduced me to the word contra to free my thinking from these arithmetic constraints.
Because of this arithmetic denotation, Grassmann introduced a rotation symbol to indicate specific rotation direction: clockwise v or anticlockwise ^. The 2 symbols are contra to each other, but this time no – symbol is used to define that fact.
This leads to unexpected results when the 2 symbols are employed in one mnemonic statement.
One intuitively expects
a ^ a = –(a v a)
But this leads to intuitive issues when
a ^ b is said to be equal to –(a v b) because the implied displacement is different even if the resultant is the same; and the rotations are contra!
The notion of conjugate process immediately appears and the identity
a ^ b = conj(a v b) suggests itself in place of the – symbol.
Thus we note that the contra rotation does not always give a contra symbol experience even if it gives the same resultant. It also highlights the need to apprehend the role of a ^ a. It is either the whole or the zero rotation: a pure rotation process with no synthesis of a product parallelogram.
Because we have ends and beginnings to these processes it is intuitive to refer to the process by these. But this leads to confusion, especially when the process is used as an operator. To avoid this symbolic confusion , a new " level" of symbols are specified. These specifications usually highlight the arbitrary nature of the ends, allowing end symbols to be specified as input and output to the process. The process then becomes a function or procedure .
As a function, removed from its defining context, the algebraic symbols mnemonic role becomes crucial to the application of the process.
Grassmann soon ran into a fundamental problem in his earliest notation. When he specified the 3 points in a plane, that were so informative in his first vision, this gave him 2 Strecken that were connected by a point of meet.. His father had specified this as lineal addition, after an arithmetic archetype or so he thought.mbut Hermann wanted it to apply to the dynamic relative rotation as seen in his subsequent meditation, and he wanted it to be simple nd intuitive. Thus he specified ehat I called the Law of 2 Strecken as always " true" . This simply meant that the sum of any 2 Strecke which meet at the ends is the configuration formed by this meet and their relative orientation. He then went on to lable this configuration with a single symbol. Gibbs simply misunderstood this detail. He took the law of 3 Strecken and completely mixed it up!
A vector sum is said to be a vector derived by adding 2 vectors nose to tail, and keeping the orientations parallel while this process is carrier out. Because he confusingly used Hamilton's term vector ( and then slipped in Hamilton's coefficient addition to justify his notion) confused students just gave in nd went with the flow!
The law of 2 Strecken only works forv3 points, and it does not give a 3 rd Strecke as a result! This would be too complicated, involving trig and Pythagoras calculations to justify!
The Law of 2 Strecken is a simple process in the 3 point case, it takes the observer on a connected journey, through the poit of meet. A to B followed by B to C.
The direction of the Strecke a(s opposed to its orientation, the 2 are not the same!) becomes crucial. It is this direction which governs the use of the – symbol. This symbol is thus a symbol of contra direction not of orientation.
Now should the observer adopt one of the points as their relative position, then the observers notion of orientation is changed accordingly, but the direction in the segment is not changed by the observers viewpoint. Direction is a displacement from a start point to a finish point, and for a specified Strecke with a specified begin nd end point this direction is the same in any orientation of the Strecke relative to the observer, or indeed any extra Strecke reference..
This was quite a limited observation by Grassmann, and was useless until he had a vision about the multiplication of Strecken!. This time his Father had specified that the 2 Strecke proceed from a common meet in a divergence . This means that the 2 Strecke move out from a single point with different relative orientations to the observer and a relative orientation to each othe. This is called diverging " treten auseinander". The multiplication concept was defined via a rectangle by Justus Grassmann, but Hermann saw the Strecken dynamically rotating relative to each other, and so generalised it to parallelograms.
The multiplication concept was the construction of a rectangle. Thus Hermann generalised it to the construction of a parallelogram.
It was when he applied his law of 2 Strecken that he noticed a beautiful similarity. The construction of Parallelograms requires the use and construction of Parallel lines. In applying the law of 2 Strecken the meet was actually constructed by a parallel line segment on the end of the beginning Strecke. This then identified the 3rd point for the other parallel line to be constructed on the end of the end strecke( ie the second Strecke). So the construction of the parallelogram involved 2 applications of the law of 2 Strecken, one in a process a + b, that is a followed by b in the sense of a synthesis or construction process. The second process was b + a . Although they reached the same point, the journeys were radically different. The configurations were in an intuitive sense contra each other. It seemed that a + b = –( b + a)
The only thing that had changed was the order of the process of construction, and this seemed to justify the use of a — symbol. In synthesis commutativity did not hold.
It went further. If he introduced a third Strecke in the plane, set so as to multiply with the first 2 in the configuration( stat) of the law of 2 Strecken, then he got a beautiful result which again showed the sum of the pieces as parallelograms made sense providing the Strecken, as factors of the construction were considered as having a fixed direction, so that the contra direction attracted the — symbol.
Now I have used the term symbol carefully throughout, but Grassmann in his early days of geometrical thinking identified it as the mathematical minus sign of negative arithmetic, ie the arithmetic of negative numbers. However he already was identifying the negative sign ith a contra direction, thus considering numbers as directed quantities, as was Wessel and others exploring the meaning of square root of –1.
Although Euler had worked extensively on the issue it still was not accepted or well respected in Grassmanns time, and Gauss in Prussia, hesitated to give a view on the question, even though he had demonstrated profound consequences of accepting these negative and imaginary quantities.
The Law of 3 Strecken also allowed Grasmmann through the rparallel line construction to move a given Strecke to any start position on the plane. Thus Strecken no longer needed to be given in a connected configuration.
What was crucially important was the relative orientation , and a further propert , divergence or convergence. If Strecken converged then the construction of the parallelogram was done " backwards" so to speak, and the result was considered as a negative parallelogram in that sense, but it's orientation was the same as the divergent Strecken parallelogram, but it was translated backwards.
There are several constructible possibilities in the plane. The orientation of the Strecken, the direction of the Strecken and the ivergence / convergence of the Strecken all factor into the description of the resultant parallelogram. Gibbs completely ignored all of these distinctions, Bill Clifford did not.
The Clifgord algebras continue this method of analysis and synthesis, as pioneered by Grassmann.
Justus Grassmann was in correspondence with a group of crystallographers who were classifying the distinctions in crystals, and developing a group algebra in the process. Hermann was able to correct a mistake his father had made in his analysis of crystals, for which he seems apologetic to his father. However, his understanding of the applicability of his method and analysis soon took him beyond the realm of geometry a la Legendre into the domain of physics and Astrology, into the domain of Newtons Principua Mathematica.
He developed the notion of the Schwerpunkt, the projective geometry through vertical projections of Strecke onto relative Strecken, and his realisation of the inner product in ratio with he outer product. These concepts were. Analogous to newtons, including his method of First and last ratios and infinite series. Newton, like Grassmann after him, and the Greeks before him, worked with magnitudes and quantities cut from magnitudes in space, he did not work with the concept of Number beyond its value as coefficients in the binomial theorem, and of course as notation for ratios in the sine and Logarithm tables. Logos analogos was fundamental to his thinking and his algebra, as it soon became for Grassmann.
These distinctions in Grassmann's method meant that we no longer had 3 dimensions or Strecken for a reference frame. We had as many as we needed to describe a space. The fact that Contra often simplified these Strecken into " negatives " of known ones , still left those configurations where there were no contras, such as the tetrahedral forms, as apprehensively by the Grassmann Algebra, and now the Clifgord Algrbras.
The Cartesian mosaic dominates, but the Grassmann analytical and synthetically method has enabled forms that are difficult in the Carteian mosaic to be apprehended in an appropriate mosaic and then transformed to the Cartesian notation if necessary.
Now the advent of software to represent the configurations means that this transformation does not need to be done. We can work directly with the Clifford/ Grassmann algebras.
The Schwerpunkt is the centre of symmetry for any form. It is the point which radiates all Strecken orientations at once for a given form. The dynamic of the form must balance as in the Barycentric eigen value eigen vector calculus. But independent of that one can take any straight edge and slide them parallel to their original orientation until they all emote from a single point. This is useful for a frame , but not useful for a solid, but it provides sn animation of how the orientations of a form are all parallel to the orientations diverging from a central poit , or a common point, in or out of the body or form. This is the basis of a reference frame, and an n imensional basis.
The only error to avoid is the insistence on all basi vectors being orthogonal! It is enough tht thy re not parallel. It turns out that if the inner product is 0 then the Strecken are orthogonal, and the outer product exists, but if the inner product is. 1 then the trecken are parallel, but the outer product does not exist! Ie it is 0. The inner product is the ratio of the inner vertically projected product to the exterior product, where the product is synthesised by a construction using parallel lines.. The inner vertically projected product results from the exterior Strecke projecting vertically onto each other to identify the inner projected Strecken., which are then used to construct a parallelogram. The ratio of the inner parallelogram to the outer parallelogram is the Inner product Bon Hermann Grassmann.