The significance of Group theory as an analogous description of field effects derives directly from a coming together of group theoreticians around the quaternion 8 group.
Euler perhaps set out a method of group description in his equivalence class theory, called Modulo arithmetics,Galois is supposed to have solved the quadratic polynomial group structure, writing it all down in scribble just before he died in a duel, and several other noted European mathematicians kicked over the traces of ring and group theory, but nothing as significant as Lobachewsky attacking Euclidean geometry so called, in Gauss opinion. Thus while Gauss responded to any encroachment on his developing geometrical ideas directly, seeing off Grassmann, Bolyai and learning Russian to tackle Lobachewsky in short order , he underrated the growing significance of group theory.
Hamilton is rightly hailed as the father of modern algebra, his contributions to group theory being significant: both the complex number group and the quaternion group, new groups in addition to the Eulerian cyclic groups the others were tinkering with, plus a substantial and consistent theoretical mathesis to underpin his work; it is in fact grassmann who single handrdly defined the foundation, structural supports and the roof structure of modern group theory. Of course his work was deliberately buried by Gauss because of the geometrical implications, but was also buried was the new superstructure of group theoretic notation Grassmann had drawn upon to describe and situate his work. The ideas are firmly seated in work Justus Grassmann had done in 1827, but also in ideas about structure and form his brother Robert had. So the overall result was a Grassmanian, Stetin, approach to the nature of reality and the place of knowledge production, or Science within it.
To describe all the potential his analysis gave Herrmann created a mere demonstration model called the lineal Algebra, notated in group theoretic language as we now use it!
It is hard to grasp and get across that the lineal algebra was only a demonstration of what could be done. There was and is much more that we can do using zfrassmanian analytical tools!
So why use Groups instead of fields? Fields we cannot see, but we can demonstrate the properties of. It just so happens that certain groups underpinning the metrons we use to measure have precisely the properties of the field. Thus we can study the group to mimic the field, as well as to mnemonic ally capture the field properties. It is very abstract to study a mathematical group instead of messing about with magnets and charge, hey some like to do it this way others that. The point is they are both analogous to each other.
Now if the quaternion 8 group is the basic field group, then it matters what it is homeomorphic to, So I am on the track of that as I write.
I have given it some thought, and have concluded that the quaternions are a mess heuristically . They can be designed more neatly and with far more properties, based on the roots of unity.
Firstly after many trials with different cyclic groups homeomorphic to the Quaternion 8 I decided that Hamilton had twisted a relationship in the cyclic groups, either by design or by force of invention! He had struggled so long for the solution that he jumped at the first sign of a break through. The fact that it was a non commutative result was irrelevant in the circumstances. The pragmatic point was that it worked.
Now I have an explanation of how it works in terms of the cyclic groups, and this releases the constraint placed on it by Hamilton allowing for future developments.
Firstly, like me Hamilton wanted to construct a 3d solution using the complex functions as rotators in the unit circle. However he did not restrict the action of the imaginaries to a single plane, so they acted in all parallel planes. We therefore construct a reference frame using 3 circular planes orthogonal to each other. This is the approach I used in polynomial rotations. It only works by brute force, as the underlying algebra is quaternion, a fact I sought to prove otherwise, but it is clear enough.
Now Hamilton did not have a fractal generator to cover over any mistakes, he had to construct the underlying algebra and that required a freedom in spatial action that I restricted myself from formally.mi only allowed the operators to act in the one plane of definition, and I had no clue what these operators were, only how I wanted them to behave. Thus my constraints determined my model and my boundaries of investigation..
I think that Hamilton decided to use 3 circular planes but how to place the third plane is the source of his inspiration.mthey were to be placed around the shaft of rotation like the flights on an arrow or a dart!
Thus the best cyclic group structure is Z4@Z4@Z4
This consists of 64 elements only some of which have to be defined carefully.
Now it is tempting to combinatorially tidy this all up and deal with only 16 pair combinations, but to do so obscures what is being done to the structure. 000, is the structural node for the axis of rotation and this can be defined as the 1 equivalent, the identity element. 222 has to be the opposite axial node which provides the systems axis of rotation, although the system can rotate about a point in any configuration. This leaves 111 and 333 undefined.
As Hamilton did not condtruct his solution in this way he would have no suspicion of this design possibility let alone flaw.
It is quite easy to pick out the elements with cyclic order 4 and to choose these to define I,j,k. The problem is heuristically what do you do next? Using 100 for i, 010 for j and 001 for k only gives some of the Hamiltonisn behaviours. It also gives several versions that have to be defined as -1, which are 200, 020, 002. The consistency is maintained because these all conjunct to make the -1 node 222.
The problem arises around how to evaluate the 16 pairs consistently. Hamilton's solution ijk = -1 does not at first sight appear to be part of the system. Ijk is 111 and we need another 111 to get to Hamilton's solution. Now I am going to define 111 as arc and 333 as – arc. Hamilton's solution uses this aspect of the structure without knowledge of it instead of defining i as 100 I define i as arc-100, j as arc-010, and finally k as arc-001 , these are conjugate definitions of the original or initial ones. These conjugate values are not at all obvious unless you study the roots of unity. They give the results Hamilton was inspired to carve on a bridge in Brougham . It also explains why the results are not cyclic or commutative in general . When conjugated are joined in a combination they alter the commutativity of the combination relative to other combinations. The commutativity in the element tables therefore does not predict the commutativity in the combination form. But if the product terms consist of these commutative elements commutativity is transferred to these product terms. Hamilton did something to break the commutative chain in his tables, and I think it is what I explained above. Effectively he equalised the conjugate terms. This was clearly done in ignorance, but the non commutativity of the quaternions is in part due to voiding all the conjugate quaternions for the k label.
The whole set can be redefined in a cyclic commutative way giving access to new possibilities.
The main outfall of this analysis is that the system both Hamilton and I are using is based around an old solution from spherical geometry: divide a globe into great circles around a single polar axis. With this solution I can transform from the tables of relations to a spherical display system, providing I have as clear polar distribution in the tables. This is not at all clear in some sets of relationships, making it confusing for me to pick out how best to display it. For thr moment cyclic groups offer the best chance of making such mappings to Cartesian or polar or even to quaternion reference frames.
In my original construction for polynomial rotations, I wanted the third plane to cross the other two like a latitude line crossing two longitude lines, but because of my assumption that the rotators acted only in one plane. I was stuck. Also I assumed there was some rotating force turning the "wheels" so to speak, I did not recognise that the underlying algebra was a programming instruction to tell any computational process how to proceed! Thus the instructions in the tables were the functions of actions, the rotations themselves!
The construction of such functions is a fascinating insight into what "Mathematics" is. It is the construction of tables with the systematic rhetorical instructions of how to use them. If this sounds like software encoding that is because that is precisely what "mathematics" is. Mathematicians have always been placed heads above everyone else since Pythagorean times, but before that Astrologer priests and philosopher sages were revered for recording (Tyme-ing) the heavenly gods and pronouncing on seasons and omens. By building up huge tabular data banks, systematic analysis revealed algebraic patterns, and these patterns were recorded rhetorically for use by those who wished to be Astrologer priests.
The Pythagorean Scholars therefore provided the first open access to such knowledge and information. This is not to say they gave public lectures on the grubbing of predicting Venus transits, even as astronomers do not do so today, but they provided access to everyone who wished to study with them the same access as students of higher education have today.munfortunately they had enemies, making it very dangerous to be as open and communal as they wished.
There is no doubt that much of what is Pythagorean developed around the Jesus mythologies and experiences, which is not to say that neither of these two people were historical characters. The evidence is that both existed as real people within their cultural settings. But much water has passed under the bridge since then, and much manipulation of reputations.
Mathematikos therefore seems to have been a Pythagorean mark of distinction for a qualified astrologer, one who could rightly proportion the heavens, and describe each planet in its course and in its season. In the meantime, Plato seems to have set up his academy on the basis that students be competent in surveying the land before they come to the academy to learn how to survey the heavens!
The richness of these pursuits lies in the mountain of data available for a student to programme his/her brain with, and to process into better and better rhetorical descriptions. Because Plato admired the immense knowledge and disciplined thinking of the astrologers, the Mathematikoi of the Pythagoreans he established his academy to promote this level of excellence in all who wish to govern for the good of mankind, for all who had a utopian ideal like the Pythagoreans.
Mathematics has never seemed to lose this platonic respect, but in fact it should now had over that mantle to Comouter science and data processing, as this has now essentially taken over that role.