Associativity, Anticommutativity and Antisymmetry in Grassmann Algebras

http://www.stebla.pwp.blueyonder.co.uk/papers/Euclid.pdf

Normans Wildlinear Algebra series is a great introduction to Grassmann algebras. But his projective geometry is also a Grassmann algebra, something Robert Grassmann particularly emphasised to Hermann in his 1862 revision of Ausdehnungslehre.

Finally the hyperbolic geometries are also Grassmann algebras.

There is an expression revealing Grassmann claimed to have realised this in 1844, and it is clear in 1862 that he had hoped for collaborative researchers to have advanced his ideas into a new root and branch formulation of the house or tree of Mathematics and the sciences. By and large. Peano and Whitehead, Gibbs and Clifford have managed to do this, with Dirac being the jewel in the crown.

I have struggled to appreciate what he was thinking, much of which is inspirational and subtle in his German originals. The notions he struggled with are well put and far reaching, as expressed in his Vorreden. One of the notions is the notion of “contra”.

Hamilton uses this notion to build his model of conjugate functions, otherwise called a science of pure time. He struggled also to break free from Cartesian coordinate thinking, and created much of the vector terminology. He acknowledged that Hermann was ahead of him in this game and redoubled his efforts to catch up.

The notion of contra underpins the sign conventions. We can start with Brahmagupta who introduced his fortunate and misfortunate numerical and algebraic and also aphoristic designations.

I can go next to Bombelli, and. Napier and Stevins for bringing it into European thought, against opposition by Hellenistic protagonists.

Then to Descartes Newton and Leibniz for making it de rigour against considerable antipathy , and then I think to the Grassmann’s and Hamilton for developing the ring and group theoretical superstructure to apprehend it.

Contra, as a notion applies not to the elements of objects things, etc, but to the subjective processing of the observer/actioner. It is that individuals point of view or perspective or even apprehension that is being modelled by contra..

Often I read some researcher stating that there is no negative numbers in our experience of reality, or space. While that seems prima facia to be a truism, in fact it is a misapprehension of such thorough obscurity that one cannot explain it to one who is of that opinion. One simply has to learn that our models are only a small eclectic choice of all possible and impossible models! And then one has to apprehend the group or ring structures of all these different possible approaches and models, and how that affects ones appreciation of any supposed difficulties.

Eventually one may learn that our models programme our abilities, behaviours and reactions and viewpoints.

On a purely utilitarian criterion, some models are more useful than others. Adding the contra notion has proved to be extremely useful in all sorts of comparisons and judgements.

Sometimes we get stuck in one way of thinking, so contra means only one thing. But usually, by analogical thinking we recognise contra in many descriptive experiences. In this blog I am looking at contra in terms of spatial orientation and spatial symmetry. Because it is a spatial comparison I need the notions of associativity and within associative relations the notion of commutativity of the elements of the relations.. It is this commutativity which invokes the analogy of symmetry, but not just as a metaphor: it is a cognitive apprehension of the spatial relationship of the elements! In other word we have to quit being objective and examine our subjective processing of the actions, notions and relations , the ennoia going on with our mental engagement at the very least.

Symmetry derives from summetria, a fundamental Greek notion of group structure! The term seems to have its origin and weight in architectural concerns, where it’s principal notion means a ” common” or” gathering” measuring scheme. Thus a Metron , a single measure is only part of a summetria. A gathering of Metrons/ Metria.

The underpinning concept of a summetria is a beautiful or harmonious form. The standard example is the human form. This is not a conceit, but an important design principle. Humans feel more comfortable if their surroundings do not jar with their familiar Metrons. Put another way, architecture has to have human dimensions!( Metria).

The analysis of what is judged aesthetically pleasing reveals a ” collection” of Metria, different scales for different regions of a form.nfor the human we have phalange size, gfnger size, hand size arm size, abdominal size head size etc. these sizes or Metria are tabulated for the aesthetic ideal. You will note Da Vincis man in a circle and square as a geometrical exemplar of this.

During the renaissance, artisans studied these summetria, that is collections of measurements to learn how to compose the aesthetically ideal forms. Such forms therefore were designated as being ” of summetria”, that is derived from symmetrical considerations and meditations.

Much, much more can be said about the proportions derived from the summetria, not the least being the golden ratio, but the thing I want to note is the fundamental model a summetria presents of a group structure.

The elements , the closure, the combinatorial actions are all exemplified in this concept. For example, any building designed by these principles will scale, so the ratios and proportions are fundamental distinguishes of the Metria within the group, and to what the combinatorial resultants must conform! This means that the group has a quadratic action on its members when deigns are considered and elements combined. The golden ratio forms give a simplified superstructure to get designs ” right”.

I use this derived notion of a group distinction process: a common bond, or binding or boundary defines the elements of a group. It also defines the essential meaning of a summetria.

So how can a group be anti symmetric? It can’t! What is anti symmetric is the action required to keep resultants within a group.

A symmetric action always produces resultants within the group. All closed groups are therefore potentially symmetric. An anti symmetric action produces resultants that require a contra element in the group. Thus a group has an extension which is contra to the group and these are combined to form a new group which now is symmetric, but only through including anti symmetric resultants. The group can be partitioned with no overlap!

The action that produces an anti symmetric result is often referred to as symmetry breaking. What this means is that sometimes our models are forced by us to be symmetrical, when the reality we live in is antithetical to that constraint. We have therefore to submit to empirical data and modify our model, no matter how beautifully aesthetic it is!

The fundamental action behind Grassmanns analysis is synthesis or construction.mthus Strecken are construction lines, points are construction points, and the combinatorial actions of synthesis simply mean draw and construct using these TYPES of element…. The construction of a line read draw a straight line using points A,B. a computer would ask for specifics, a smart computer would access the types and use the default instances. An even smarter computer would ask for specific properties to be input to alter the default. This is how Grassmann synthesis works, but we must know the default construction to use it!

Before we can start to synthesise we must have analysed the “finished” or goal form. When we have that basic knowledge as default information we can use Grassman’s synthesis algebras. His algebras only have to specify the elements and how they combine. The first aspect of combination is ” pros” that is placing in a relationship or association by putting an element relatively befor another element.

In space no one can hear you scream: “what is ‘before’ something else!” the answer is not spatial but sequential, and it is sequential in process. So when I describe a as bing before b I can interpret that spatially only if I establish a sequence of processing. The spatial notion of “before” or ” pros” derives from this sequential action. This sequencing is totally subjective.

From this notion we also derive the temporal notion of before in time. In fact Hamilton’s synthesis of algebraic arithmetic is based purely on this notion of ” progressive” sequences, which he identifies with pure time .

So associativity is about sequencing the elements. Because we start with 2 elements we mistake this point. The 2 elements are associated in a sequence. Because of this we can actually have as many elements as we want in an associative sequence.

Commutativity comes when we commute or change two elements in the sequence. In fact it is any two elements in an associative sequence whether adjacent or at different places in the sequence. Most Group structures follow Euclid and deal only with the two sequence and the three sequence. When we look at commutativity in the three sequence we introduce brackets to emphasise what are commuting. The bracket creates a new object, it is not an element it is a synthetic structure made up of several elements in associative sequence. Beyond that the bracket is left unspecified.

However we can now isolate which elements we want to direct our focus onto. These elements cn then be synthesised indecently, and then the resultants combined in a synthesis. This creates a problem. Because if my combinatorial action is specified on points say and produces lines, can my action apply to lines?

The answer is no if you take an anal approach and yes if you take a goal oriented approach. If the goal is to construct a parallelogram, then we must use the points and the synthetic products of points to do just that!

Anally this is horrible, because as a computer we have not given it enough information! It takes a demonstration of what to do, what to vary etc to specify the construction! However for a trained monkey this is quite doable. The difference is that the monkey has analysis or analytical skills to help it learn nd try heuristic ally to reach the goal for a banana!

The use of these additional subjective processes in this way are what mathematics and science had ignored until the Grassmanns. They did not exclude the observer from their prt in the construction process.

Effectively the notation takes on the role of human software, or a set of construction instructions like those found in an IKEA flat pack!

Now, we can make sense of Antisymmetry and anticommutativity. They represent the very real differences that occur when you attempt to construct something in the real world. If A and B are processes then sequencing the processes AB does not give the same result as sequencing them BA in general . It gets worse! If a,b are elements in process A then regardless of elements in process B changing the sequence of elements in process A will change the resultant of AB in general.

Thus we se that the commutation of associative sequences is fundamentally anticommutative .

This is not true of representatives however. Representatives can always be commuted. This gives us a general distinction between representative terminologies and fundamental constructive element terminologies.

In general associativity as defined is not going to necessarily give the same resultants, because the bracketing actually implies a different process order within an unchanged sequence order. The reverse polish notation highlights this very effectively, because it shows how the brackets actually change the process sequence in a computation. The BODMAS rules are an attempt to “code” between the two, so as to retain some aesthetically pleasing notation. This is why Grassmann analysis was so fundamental. Educated people were putting aesthetic form before constructive function and getting everyone in a constructional muddle.

It was not easy for the Grassmanns to see through this aesthetic fog, and Justus made some fundamental mistakes because he could not see where he needed to go! But by continued diligent effort and the rise of computer coding we have made the breakthrough, and the Aesthetes have even started to pretty it all up again!

Grassmann defined AB = –BA because he found that the sign portrayed this difference in orientation symmetry.. The difference in orientation is subtle . It is not as simple as negative or opposite. It involves not only orientation, but also rotation of the observers point of view! It is non intuitive therefore, and took a while to apprehend.

Hamilton had given the sign more thought than Grassmann and came up with he word contra to deal with its subtleties. However, by the same token Hermann explicitly laid out its subtle effects but used the same word Zeichenwechsel to convey its subtleties. So the notion was lost between the two. Lost in translation. Both Hamilton and Grassmann are not referring to the rules of sign, they are referring to behaviour from which we derive the rule of sign. Thi behaviour is that of the trigonometric ratios of the radius of a circle as it is projected onto the diameter of the circle.

But contra also refers to reversal of process as well as reversal of orientation, so the changing of “sign ” is not just about flipping from one sign to another, it is more complex than that! The reversal of sign is about the orientation of a form in 3d space relative to an observer.. While there are no negative measurements in space, there are oriented measurements. Sign is the rudimentary mark we use to alert us to orientation in space, both ours and the objects relative to each other.

The anti commutativity and anti symmetry in Grassmanns algebra relies on one fundamental process. Designation.

Designation is the same as distinguishing, so if you have two points you can designate the first one A or B, which determines the second one .mthis automatically means that A,B is not not distinguished as a pair!. However, the relationship between the 2 designations is that Streke AB = –{BA} Streke.

Most of us miss this distinction, because we are waffled around it by our teachers. The sequence position is sacrosanct so if I associate 2 points in the sequence 1,2, then designating them A,B is different to designating them B,A. Thus when Grassmann writes in the vorrede to 1844 that he had been meditating on the negative and noted that the Streke AB is the negative of The Streke BA, he does not mean that we are looking at a fixed Streke in space. The Streke AB becomes the Streke BA very simply. We redesignate the points. But this simple redesignation has a profound spatial effect; it reorients the observer! Alternatively it flips the line round ! It does this either in the plane or out of the plane, around a centre of symmetry placed anywhere in space. In fact the flip can be specified, but it is usually left unspecified?

This is obscured by the modern vector treatment of Grassmann. A point is designated by the observer. Justus set some ground rules. Points should be designated in alphabetical sequence so 1st 2nd 3rd maps onto A, B, C, and direction follows alphabetical order, and construction is done in the same order, and crucially angle measure is done in the same order!

Changing one of these rules changed the resultant construction. Everything was specified, the only free thing was the placing of the first 2 points and the drawing of the lines. Points therefore had no specific position on the page and a line had no specific orientation or direction or length until it was designated. This by cycling the designations different constructions could.be done.

Suppose for points A,B,C two lines are produced . Designate AB as a and BC as b then producing a parallelogram from a and from b is the same as producing it from b and a. However AB means more than that. It specifies the sequence of construction , the position of the elements relative to each other, the orientation between them, the direction the line is drawn, the direction the angle is measure in construction . BA therefore contains a specific instruction to put b in the first position. That first position has an orientation relative to the observer. Thus b has to be put in this orrentation, and thus flips the construction, because the angle construction has to be performed in a certain direction, that is clockwise or counterclockwise. Justus specified all angle measure construction to. E done counterclockwise. We still have that tradition, but it is the missed out elementary factor in the explanation of anti commutativity and antisymmetry.

So the notation AB means construct a parallelogram using a,b with b rotated counterclockwise from a. Thus ba dictates that a is rotated counterclockwise from b. most vector treatments ignore this fundamental instruction. Instead they place before the student the bald axiom AB = –BA and do a bit of sophisticated algebraic manipulation to produce the result.

Grassmann was actually stunned for a long time when he arrived at this result. He was stunned because the notation was telling him that he was missing something in his understanding about space. It was simply that synthesis or construction has a sequence and a specification. To achieve a consistent resultant everything must be specified. When this is done as Justus had insisted throughout the primary schools in Sczeczin(Stetin) you reveal a fundamental process superstructure which Grassmann felt was embedded in 3d space. It shows itself over and over, because in constructing the parallelogram the same anti commutativity stands out! It stands out because of the precise, rigorous, anal., autistic specification of everything! Thus I think it is a consequence of our subjective processing rather than space. Grassmann’s ” Lebe und ” Seele” metaphor is in my opinion the actual foundation of this curious antisymmetry and anticommutativity.

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Without this superstructure most of the anti commutativity notion collapses, and with it antisymmetry.