Polynomials 2!

Before polynomials there were multinomials! And multinomials were the kingdom of Newton and De Moivre! Both these men had such an astonishing facility with them, that the Royal Society in England admitted De Moivre into its fellowship on the strength of his remarkable achievements! Both DeMoivre and Newton chuckled, because clearly no one had closely read Newton's Principiae in which he lays out the theory for such things! De moivre was merely following the instruction of his Mentor, Newton!

However, Newton and De Moivre moved the binomial and tri nominal fumblings of other mathematicians into the extraordinary nth degree formulations, on the basis and notation of Wallis's work. They called them Multinomials, but by Gauss time they had ontologically morphed into " polynomials".

I have delineated how polynomials inhere all the combinatorial theory we ever use and suggest more besides. However the source of these combinatorial regulations is our own personal, subjective philosophy and Metronic pragmatism. I in effect take the underlying notions of conjugacy an adjugacy and make a process model that works for metical spaces, and the main tautological modification is to the " law" of distributivity of conjugation over adjugates in the other conjugate. This underpins the fundamental sequence notions of sequencing and bracketing, analysis and synthesis..

The restriction on commutativity is a further modelling control, but to be fair it arose out of some bad notational choices and some misunderstandings of the underlying conjugate relations. In particular, I have shown that Hamilton forced his Quaternions into non commutativity because he did not understand their underlying tructures. Nobody did! So all credit is due to him for revealing them and making them work, albeit in a bastardised form!

I have recently used that analysis to create Hamilton's triples as a functioning reference frame, and in coming to this insight on polynomials I worked out the division formulae for the triples. So in group and ring theoretic terms I believe I can shoe Newton's Triples are a field with conjugate rciprocals and adjugat and conjugate identities..

It was while using some algebraic software to cut down the combinatorial exploration, tht I realised the polynomial nature of my exploration, and how it derives directly from conjugacy and adjugacy and links directly to so called complex reference frames, or as I prefer complex vector algebras.

The link with the fundamental application of conjugacy in sequence theory was also crucial because the iterative reality of sequencing is our chief state of interaction with space and with Shunya!

Although the attempted formalism between sequencing orientation and array dimension is vague and arcane, there is a powerful organising principle behind the attempt. The first that I noticed was the row, then the row-column array, and then the row- array stack or cuboid array. But then the dimensional link breaks down!

It should break down because the dimensional packing is in our real world, and this kind of crystalline structure really only has the three orientations in which to expand. This array structure I call the right gnomon pattern. The right gnomon is a fundamental standardisation of spatial form developed from the Arithmoi of a particular era. General Arithmoi can be nets a of any pattern of schema , very similar to topological nets.

The way this particular scheme would develop is by fractal dimensioning. Thus after the cuboid array we would scale up fractally to the row of cuboid arrays, then the row-column of cuboid arrays that is the array of cuboid arrays, and finally the row-array stack of cuboid arrays, that is the cuboid array of cuboid arrays. The fractal patterning continues in this vein.

Needless to say, this structure is a polynomial one. Using this structure I was able to analyse the quaternion8 group efficiently without writing every single conjunction out, and thus to determine where Hamilton had forced the group into an uncharacteristic conjunction, removing commutativity from the group. I also used the fundamental cyclic group structure theorem to do various analyses of homeomorphic group identification. This looked and felt like polynomial comparisons.

One might have remarked how similar it was to vector arithmetic, if that was not too naive a comment! Vectors derived from the polynomial heritage in Europe, particularly in Gauss's notational form.

The right gnomic pattern was not the only fundamental pattern from antiquity. The cubed gnomon, called a lune was a more ancient system of proportional relations involving circles and spheres.

The organisation of a sequence using this pattern tends to be more organic in feel, and yet it has the same polynomial structure. However, the curved surfaces allow for orientation adjustments which lead to tree like or plant like bundles of structure. So instead of rows and columns we have branches and nodes. The fractal pattern tends to be branch-node, but each branch is a definite spatial orientation. When restricted to paper, the layout soon becomes unsatisfactory for the confinement of the paper, and yet it is The most natural representation of sequencing alternatives in 3d space.,

We have to recognise the impact printing costs have had on the representation of mathematical forms, and not unnecessarily value a format because it was how we were taught! Publishing can now more or less present a topic interactively and dynamically in the best way to grasp a subject.

So the curved gnomon gives the sme polynomial forms, but of course just the coefficients are modified.

Thus, without any fuss, the combinatorial possibilities of sequencing a given collection of elements can be laid out as a tree or vein like system, the very pattern Euclid displays in his Stoikeioon, or alternatively in a cuboid or crystal like system, a pattern that Justus Grassmann was very interested in at yhe beginning of the 19th century. There are no other fundamental combinatorial patterns. Both are fractal, and both are 3dimensional or space filling.

These simple observations have profound consequences for the underlying structure of "reality". Whatever it may be, polynomials are our only way of tapping into it. Consequently, Newton, Justus Grassmann and others felt that this combinatorial structure precedes magnitude, quantity and thus Medtronic numeral accounting, or metric space! To describe "reality", or rather Der Raum, 3d space, combinatorial theory and numeral theory would have to be combined into a Verbindungslehre, a combined theory. However, this notion of Verbindungslehre is the very essential notion of a group, the Greek summetria. So Justus Grassmann was proposing an early form of what later became known as K├Ârper theory that is body theory or collection theory.

The nature of Justus Zahlenlehre I will soon explore, but the notion of a continuous extension fifer entitled by a metron into distinguished quanta, diskrete or discreet forms seems to be his notion of the source of Zahl or Anzahl, the tally or accounting label we attach to quantity. Since it is a label, the fundamental reality is the sequencing or combination of these discrete or distinguished parts. Thus our numerals need to be distinguished by these combinatorial realities. All that means is that like is combined with like, and the numeral displays the count or tally of the like things while the likeness is distinguished by using any other symbol in the printers block! Directly this is algebra in its classical form!

Thus algebra was to be put to use as the basis for describing reality both in its continuous and it's discreet forms, or rather in its like and unlike forms, with like things being continuously extended, but unlike things being distinguished as discreet.

I will spend more time on this in another post, but this is really usually the very first brief lesson in Algebra. However, it is never more than a brief lesson, and then it is never touched on again! Well hello Hermann and Justus Grassmann! They claimed that this was a fundamental of real analysis ignored by higher mathematics. But in fact it was obscured by the blinkered view that calculus was groundless without a definition of number. Once that had been "sorted" mathematicians went back to their habitual practices, unaware of the yawning chasm in their description of reality!

Robert Grassmann took this insight into his theory of Form, and his scientific basis to Philosophy. Thus he too attempted to provide an empirical basis to philosophy by rigorously analysing it's major branches and revealing its weaknesses, and proposing scientific alternatives. Formenlehre was his approach to the Mathematical sciences based on Geometry and the interaction with Der Raum. Justus Grassmann had already proposed a Rumnlehre, theory of 3 d space by this time, o Robert was very much continuing and extending his work. Others too had worked in the area of Raumenlhre, and so it was a growing field of analytical investigation. Robert Grassmann made an important contribution to it on the back of the work of his father and brother Herrmann, it was as a publicist and promoter for them that he contributed most to the continued interest in their work. His philosophical contribution has been sidelined by the greater influence of other Prussian and Germanic philosophers, of which there were many!

The use of this basic algebra to describe space lead to new problems. For example, the polynomial theory was for one unknown quantity of magnitude. This quantity of magnitude was considered continuous, and therefore VARIABLE. When I first learned algebra, these were called interchangeably unknowns and variable. I did not have a clue why there eas so much uncertainty about what they were!

Very much later I remember being introduced to polynomials in 2 variables c,y with the warning that these were very much harder to solve! It was a kind of limbering up exercise for tackling differential and partial differential equations at A level. I do not recall ever doing much on partial differentials after that, possibly that was due to my choice of courses, but I did meet the strange Laplacian action, some tensors, and a lot of group and ring theory which frankly was incomprehensible! By the third year I was a quivering wreck lost in the book about foundations by David Hilbert, wondering about the meaning of life, God, and the imminent return of Christ! Good times, good times! Lol!

Thus these continuous quantities of magnitude, formed only a narrow part of the reality one could describe mathematically. This went unquestioned because nobody pointed it out! Combinatorial theory as ugh was never discussed outside of group or ring theory, and there it was presented as an axiom of the system or group structure! What the hell! No wonder I never knew what was going on!

One day I said to my tutor that I did not find Cantors proof of the uncountable infinity convincing. I do remember that a lot of nervous lecturers and tutors spent a whole term trying to convince me it was correct! In the end they never succeeded, but I did not know what I was onto back then, so I acquiesced after a visiting Irish lecturer tried to convince me using a proof base on the isomorphism between the reals and the rattionals. I had only just been taught function theory for goodness sake! Of course I was going to assent!

I laugh now, but looking back I think I was granted my pass degree because of all this deep research and challenging I did. I certainly know that a few comments at the time seemed to guest I had given the maths department a good kicking! Lol!

Nervous breakdowns are not so bad when you do not know what they are, and when you are young and autistic and basically programmed to build an adult life. All I know, and am grateful for is thst my religious friends who shared my life for the next 13 years enabled me to go through one of the most difficult periods of my life with hope, friendship and dignity, and a gol beyond my mere existence.

As my mother said" I was more than fortunate, I was blessed."

The ceaseless computation of the void is not just a phrase for me. It is the inner workings of my being as well as the outer projection of my reality. As such I have, in the end come to be true to myself as constructed by my life's experiences, changing everyday with the changing environment. What I think today my be what I correct or dump tomorrow. What I have written in the past, was what I thought back then, not what I think right now. If I hold onto anything it is my memories of people and events I have interacted with and what Wissen or Wisdom I may suck from that experience for what I want or need to do right now!

I frequently dump my thoughts, because if I do not download them I become dysfunctional. I am barely functional now, but my wife and friends and my friend orient me to actions that have some little impact on their world. That is enough for me, to be useful in some small way. And mathematics? Ah well that was what I mistakenly called this ceaseless computation of the void! Lol!

The Grassmann's have provided me with an endless opportunity to ferret out those quirks in my life that were placed there by the fact tha the reality I barely apprehend is never straight, always twisting and turning, and always changing! Out of that ceaseless motion I have come to be, I am and I will go on to be transformed into what I do not know, but it will be some polynomial of the building blocks of which I am currently made, and it will be some polynomial of the software I currently operate on, and that will be the combinatorial connection I will have ith this spatial void which is not empty, not still and is timeless.

The combinatorial actions are sequencing and bracketing. The combinatorial properties or rather outcomes can be represented by tree or array diagrams. These diagrams combine the notion of sequencing and bracketing into chunk sizes. These chunks are the nomial terms in a combinatorial polynomial which synthesises the space or reality of these sequence elements as a combinatorial conjunction of their relationships. In particular, should thst conjunction be derived by conjugation and or adjugation, then a fundamental or foundational description of all the combinatorial possibilities in that space has been arrived at. All that is needed is an effective and transparent terminology to unlock the pragmatic modelling capabilities of that distinguished space.

To say that we have come to a deeper understanding of reality is to mislead greatly! But to say we have an analytical tool which Nablus us to construct viable and utilitarian models of our experiences of reality is accurate and more profound! For if I understood gravity in terms of us drastic equations, but now find that quartic equations give me a greater range of conformity to my experiences of it, then yes I have extended my apprehension of space and my expectation of it. But this has come about because I have a way of building a model of gravity that utilises the correct collection of parameters to describe the motions, or rather the corrected collection of parameters. That this combinatorially means I reach a quartic equation to describe the motion is significant combinatorially, but only if the sequencing action and the bracketing action is understood.
On a wider note, if sequencing involves repetition, then that produces a different possibility space to sequencing that is restricted by options, that is has a restricted set of degrees of freedom. Thus if we know which sequencing action is involved, then we can determine the significance of the results! So in a restricted sequencing setting the fact that 4 parameters are necessary and sufficient and maximal is highly significant for a quartic equation result!. But less so if the description is that 4 parameters are the minimal necessary nd sufficient set. Arriving at a quartic equation is interesting, but only one of many possibly other equations of higher degree explaining the experience.

In this characterisation I an using the combinatorial algebra ith the Lagrangan constraints to determine the significance of equated results on mpirical data. I also am using an underlying sequence theory to locate the significance in the sequencing action.

That a polynomial should arise out of such considerations may seem surprising. But I hope by now I have covered enough of the fundamental mental processing that derives or arrives at these polynomial syntheses .

Finally my general combinatorial structure for " Mathematics"
Starting with a right triangle in a circle as a tool for metricating a proportion scape of forms, I analyse each dimension into a distinguished trig ratio. The trig ratios are distinguished by labels which encode orientation, for example I,j,k.
These terms are then combined to describe a space in what is called a linear combination, but which I have identified as one of many bracket actions combining sequences.
The rules for expanding these brackets have to be specified , and that is done using the polynomial algebra. This gives distinguished sequences as terminological descriptions of the original proportion scape. These sequences have to be identified within the proportion scape so that a judgement can be made about whether this particular process is utilitarian. This usually means that as a group action it is closed, producing results that are already within the proportion scape.

When this is the case we may say that the polynomial algebra defined in this particular way is a utilitarian model of that space action. Next we can do all our investigstions in terms of the Lagrangian constraints on the polynomial algebra, and see if that again results in any useful descriptions of our experience. If this does happen to be the result, we may then use the model with the Lagrangian to explore our "reality".

Now this would be overtly a mathematical exploration were it not for computers and computer graphics and computer models. By using the method outlined above we can now create model realities in which every point or object sits within a Lagrangian constraint. By then running the programme for different changes in those constraints we can calculate the iterative effect s on the model, and display the results. In addition, our computers are fast enough to do this in real time!

Computer modelling is now a very effective and utilitarian tool where we have developed the correct Lagrangian constraints. Thus we can model air flw for many expected conditions in flight, but not for unexpected ones! Of course we could randomise the constraint variation, but ths would be of little pragmatic value. Instead we take data samples of actual conditions that occur and we can then display them on the system. This then becomes part of the expected flying conditions and we can train pilots accordingly.

The random event is always going to occur, but if we get enough empirical data, we can model its frequency and nature. So for another example, the freak wave that appears out of nowhere. This was actually a computational result well within the capabilities of current models to generate. However, because the prevailing experience of the seas did not warrant pushing the parameters beyond certain ranges, it was never seen on modelling runs. It took an investigator willing to explore the anecdote of survivors to find these freak waves in the model, and then to find them in the ocean by satellite observation!

Thus this approach by the Grassmann's has direct applicability to our real experiences, and represents a viable computational modelling and analytical tool for describing my reality. This is why Grassmann is studied today. It is not that we have not come up with workarounds, it is because a consistent theory has been proposed and for all we know may be the best theory ever. Certainly the current theories have derived from its influence much I take for granted in this modern world. What if we studied and applied it properly and fully? What then?


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