Kujonai introduced me to a notion of signs which he developed as polysigns, but I think the name polysigns was coined by ted. However, kujonai had developed his notion on the basis of the roots of unity, Demoivre Cotes formulation, and further he had introduced a modulo arithmetic description of the action. For a brief while we corresponded at Fractalforums.com, and then we went our own ways.

The issue was how to develop an algebra for the triple(x,y,z) and we all thrashed about with innovative ideas, and had fun! Because of Kujonai I attempted to develop the idea of polynomial rotations. It seemed promising, but soon taxed me to my limit! It was because of it that I came upon the notion of the roots of unity. As I searched for a deeper understanding of the problematic I notation.

It has ultimately been a rewarding experience, finally connecting me back to the very foundations of mathematics, the mysteries of Astrology! Of course I do not mean the fortune telling predictive side of astrology, yet even this has it's deeper value, and finds a noble place stripped back, at the heart of modern science. I mean the philosophical interaction with space and all it's inhabitants.

So why would i have anything to do with astrology? Simply the unit sphere, and it's counterpart the unit circle are the fundamental magnitudes which I measures. The ratio of any arc to the radius or diameter to the sphere or circle is measured in the standard quantity called i. We might consider the ratio to be a ratio of lengths, but it is in fact a ratio of divisions of the periphery of any great circle, and the standard division called the right angle, is the one used to define i. Because no length was assigned to this division, the ratio is not one of length, but it is a pure ratio of quantities, the right angle to the radius. It was called i by Euler, but it was Cotes who distinguished it some decades earlier in association with the curious process of determining the square root of -1.

The square root of a negative quantity arose from the formulation of general methods of determining square roots. Such methods relied on Euclid's use of the gnomon, a system of proportions. The gnomon, as a system of proportions was in fact a special method of the more general system of proportions based on the circle and ultimately derived from the systems of proportions in the spheres. Thus by utilising a special system, the relation to the general system was lost. This spherical system encompasses all quantities and all directions, and the loss of the element of direction made the interpretation of the negative values that more difficult and mysterious. However, we did not lose the glint of interest or intrigue, and we stumbled across the solution many times without realisation.

The process of finding roots hid the process of finding geometric means from view. The geometric mean is the very foundation of the gnomon proportion system as it sits within its theoretical base within the circle as crossing chords. The introduction of direction rightly changes nothing in this method, but it does change our notation and interpretation of such notation. The – sign has no significance if it is not a direction indicator in this context. The _/ sign has no significance if it is not a function or procedure call to use the Euclidean method of finding the geometric mean, and this method relies upon the arc of the hemicircle. This constant arc is of course proportioned by the diameter, and the ratio of this arc to the diameter is tat of the quarter arc to the radius. This ratio is i.

Now of course one is immediately drawn to the ratio pi, but this ratio, though clearly related, differs by being a ratio of a magnitude called length. Thus the same magnitude is compared in pi, but different magnitudes in i for example degrees to radius has the ratio I, but radians to radius pi.

The fact that we should have such a ratio is not at all strange for we have many such ratios and proportions to which we have not given such names, but names such as judgement, or wisdom, or even equity. The general Greek term for this is Kairos. As humans we proportion all things, and within our proportions we encapsulate the fractal universe!

The modulo arithmetics is one of Euler's more powerful methods of analysis and synthesis, but of course it originates in the Eudoxian proportional system in books 6&7 in the Stoikeioon of Euclid. Not only are the arithmoi generally misunderstood , so are the concepts based on them . Uclid describes a relationship called the common measure, and he describes an algorithm to find it. It is what is left over after this algorithm has been carried out that is the subject of modulo arithmetics. One has to realise that the terms Artois and perissos derive from this algorithm and what remains. Perissos is what remains if the algorithm does not produce a common measure, Artois is the term when the algorithm does produce a common measure. Now the common measure is called the sunthesis that is the highest common measure out of which any commensurable arithmoi can be constructed as pollapleisios , . Thus commensurability is this exact or Artois common measure, not the perissos or inexact, approximate measure.

Perissos is a measure or arithmos that can range from one monad upwards, but never more than the largest common measure. Now unfortunately these terms have been translated as odd and even., instead of inexact and exact, or even approximate and accurate. The remainder is always perissos, but some perissos are also protoarithmos, that is a prototype arithmos. Such perissos easily distinguish themselves as being uncommon measures! In fact you have to use them as a measure to build the arithmoi they fit. In this exact sense then they are proto arithmos, tpes of possible arithmoi.

Now there are lots of other distinctions Euduxus through Euclid makes about thes arithmoi in regard to the algorithm for commensurability, but all that concerns me here is that the perissos are the remainders of this algorithm, and thus the algorithm is precisely the modulo algorithm of Euler.

The power of this modulo algorithm can be shown in the demonstration of in commensurability of the hypotenuse of a right triangle. If we reduce all quantities to remainders mod 2, that is to say all arithmoi that can be constructed from 2 monads are distinguished. We are then left with all arithmoi that have 1 more monad than 2. Now Pythagoras says that the arithmos on the shorter sides of the right triangle sum to that on the hypotenuse. Thus if the hypotenuse is 0 mod 2 then the two shorter sides must be 1 mod 2.; or if the hypotenuse is 1 mod 2 then the other 2 sides must be 0 and 1 mod 2. Working through the different cases you find that not every hypotenuse can be a square made up of commensurable arithmoi. The simplest case is the right triangle with unit squares. The square on the hypotenuse must be 0 mod 2, but you cannot make any square with 2 monads! Nor can you fit 2 monads on that side.

The 2 monads have to be altered to fit, thus destroying their monad status, and initiating a hunt for a smaller metron to use as a monad. However, the very same analysis applies, and if these unit monads are replaced, it must be by a metron which makes them both 0 mod 2 or both 1 mod 2. But again no square can be formed by combining 2 equal squares because suh a square must have a semi perimeter made of equal counts of these new monads and the only way that can be done is by placing them about 2 edges of one of the squares, and this process is precisely the algorithm of commensurability. Thus assuming Pythagoras to be correct we must find a common measure by this process. However, we do not.

Repeating this process ad infinitude gives us better and better approximations(perissos)but no artios.

We can consummate the demonstration by observing that every pair of squares either 0 md2 or 1 mod 2 can only give us a square on the hypotenuse when the semi perimeter has the correct relationship with the count of monads, that is they are commensurable.

Now, this is usually portrayed as a big finding for the Pythagoreans, but in fact it was not, because in commensurability was a common occurrence. What was troublesome was that there was no certain unit to every arithmos, and in fact the monad could be of any choosing, and involve one in indefinite processes. How Eudoxus solves the difficulty is by clarifying proportion, or Kairos. The proportion of things once determined meant that there was a definite unchangeable relation. Thus a truth was definite in its relation at all scales, and the same arithmoi applied to large or small scales of things. It was Democritus and Leucippius who determined that a principle of exhaustion would in the nd bring us to the indivisible atom, the unit of all hings, making all things commensurable. But of course the argument fails because of Pythagoras theorem. The dispute was therefore over his theorem, whether in fact it could be true! Of all things Pythagorean this is indeed held to be eternally true. No one questions it today, for it has so many demonstrations of its veracity, but apparently it was questioned in a leadership contest among the pythagoreans, and we have only snippets of the outfall of that division. What did survive to this day is the absolute faith in Pythagoras theorem.

So we return to the roots of unity and the modulo arithmetic. I have often stated that there is no basis o unity, by which I mean that a unit of any description is utterly and totally arbitrary. However I am now going to discuss how the roots of unity fom a vector basis for the whole of space, and in so roiling utilise kujonais idea of a modulo arithmetic.

We start with . u^{}_{n} as the nth root of unity. This means that u^{n}_{n} = 1

It also means there are n roots of unity that are formed in this multiplicative sequence, each of which is distinct from the others, but not independent multiplicatively. We can form a polynomial of degree n of these roots of unity, which obey the algebraic rules of polynomials. Thus, though not independent basis elements in the sense that there is no relationship between them, they are independent as terms in a polynomial are independent.

One of the main reason for independence in basis terms is to ensure that no term is a simple multiple of another, so that those 2 terms are in fact commensurable in a simple multiple sense, ie they are scalars of one another. The terms in a polynomial have a exponential or logarithmic relationship, sometimes termed geometrical, and as such are not collapsible arithmetically or algebraically. This is the sufficient distinction of independence that I will use in this case, and it is a necessary one for a basis.

Now we have to understand that until Hamilton, roots of unity were determined in the plane in which a unit circle coud be drawn and a polar coordinate system established to represent complex magnitudes. Thus no one had any idea how to extend it to three dimensions, in free space, nor indeed what that might mean. Hamilton's quaternions are thus really definitive of roots of unity in free space, that is roots of unity in the unit sphere!

So, proceeding as Hamilton did we are going to establish the polynomial

P(u^{}_{n}) = Sum(a_{i}u^{i}_{n}) where i =1,2,3,…,n so thst u^{i}_{n} are all the distinct roots written out as powers of the nth root of unity, and the coefficients are distinct.

Now each of these terms, the coefficients with the root has to be considered as 2 things: first and foremost a label for a distinct orientation in space, and a separable vector in an n dimensional description, which can only be written in its n dimensional form. Thus the entire basis is made up of the vector sum of these n dimensional vectors.

If you were in Hamilton's shoes you would not have this insight to guide your thinking, but fortunately Hamilton had some kind of notion of this, but it took 10 years to gel!

Now we have to define and determine the algebra of these vectors, given the Lagrangian constraints between the independent labels. Because the labels are fully written out as roots of unity, we can inspect the Lagrangian constraints in situ for the terms. We then have to investigate associativity for 2,3,… Labels conjuncted together, determining the interrelationships of the labels under the underlying multiplication of the roots of unity.

This really is for me the most fundamental insight into the fractal foundations of Manipume, the revelation of the children of Shunya, in their ranks, their tribes, there families, their nations and in their universes.. To think that Cotes was so close to writing this all down in discussions with Sir Isaac Newton, the proper father of it all. For both Cotes and DeMoivre were students of Newton, who were profound students, not mere lip servants. And because of this was granted also to them some of the flame of his genius, which to others less diligent was hidden away. Truely. Newton's visions were ineffable, and inexpressible even by the most cogent of words, but to be able to sit with him, converse and correspond with him was the most important experience in understanding his conceptions.

There is no doubt that Newton understood himself, however unworthy, to be a servant of his God, a servant whose role was to write down the fundamental ordinances of God with regard to the mechanical functioning of His creation. That he took this seriously is testified by the course and conduct of his life. And when unfortunately his clear vision was removed probably by syphilis or some degenerative disease of his ag, maybe even by the ravages of the black death in their final throes, he desperately searched the bible for guidance back to the true path. He clearly found some resolution to this state of affairs, but that genius mind was shattered, he knew that. Much of his insights are in his copious scribblings in his notes and scrap books, but that which was his essential state of genius mind was gone from hs consciousness. He could not recall certain flashes of inspiration or sources of innovation in his latter years. We have to reconstruct from his notes and the writing of others, but through Cotes and DeMoivre we may understand the extraordinary harmony of his thinking and genius while he had it.

The binomial theorem and series, the infinite logarithmic series, the methods of exhaustion applied to differentials, the dynamic method of fluents, the theory of Fluxions, the effusion of Euclidean philosophy of forms and ideas, the calculus of the sine ratio and the formulation of its difference calculus, the difference formulations for compound interest and the binomial theorem thereto, the extension from quintic equations to multinomialists equations, in preparation for his theory of infinite nomials or the infinite series expansions derived through the method of exhaustion, by now ready to be called the method of fluents, the basis of His differential calculus, the projective geometrical solution to Mercators logarithmic series refining mercators projective geometrical ideas based on Pappus theorem,the secrets of trigonometry Indian style according to the Naperian methods, in which the fundamental secret of of Bombelli's vector and Napier's vector all lie in harmony. The simplest intuitive insight into De Fermat's theorem/ conjecture.

To hold all this insight in one mind is clearly phenomenal, but add to that the ability to compute within a corner of ones brain exceptionally iterative and long calculations and you have a giant of ability, housed within the most timid and humble of men. However, add to this his undoubted Autistic Spectrum Disorder and one has a measure of his complex ability and limitations.

While I write u^{}_{n} one really has to realise that this is a label for a factorisation De Moivre performed on a ratio combination derived from the unit circle and the application of Pythagoras theorem it says beautifully within the encompass of one equation, that if Pythagoras theorem is true then we have only the sum of the squares of trigonometric RATIOS asthe unchangeable basis of any conception of unity!

Cos^{2}( arc) + Sin^{2}(arc) = 1

This can be factorised De Moivre realised using the constant i

{Cos (arc) + isin(arc)}{cos(arc) – isin(arc)} = 1

And this says that unity is structured as a system of factors consisting of ratios of magnitudes of space relative to the unit sphere/circle.

It was Cotes who in love with Napier's logarithms, Newton's differential methods, including the binomial series , it was he who differentiating using these factors, along with De moivre, fully established the Cotes De Moivre theorem about thes factors. It was Cotes who realised that they were a unifying relationship if only because they were the factors of 1, and it was Cotes who suspected that I was a constant arc within this identity.

Although we call Napier's logarithms base e logarithms, this is a honorific. Napier's logarithms were far more useful to astrologers than to commerce, for whom they were designed. They were based on the sine ratio, not the binomial ratio. The two in derivation are strikingly similar the difference being a subtraction of a small quantity to one, while the other is an addition of a small quantity to one: just inside and just outside the unit sphere/circle. These quantities were used as the base the arithmos for the ratios, the logos, obtained by conjuncting these bases as factors. Thus an extensive ratio factorisation lies at the heart of the logarithms, and this factorisation spatially divides into proportions of decreasing size, in the case of the sine ratios, and increasing size in the case of the binomial ratio . Thus the logarithms encode a proportion structure of space that is phenomenally geometric. The Indian system of writing numbers in this p-adic geometrical form was first extensively adopted by Napier and other merchants and travellers, but for Napier it was a remarkable demonstration of the principles of his logarithmic calculus.

Napier deserves the credit for his invention, because even the Vedic masters say so. They had not discoverd this method in all their devotion to numbers and modular arithmetics, which by the way is the source of Euler's inspiration.

So the proportioning and partitioning of space by the logarithms of Napier clearly begged for a constant. The constant for sine was 0, for cosine was 1 and for the binomial it turned out to be incommensurable. That however did not stop Cotes and Newton calculating this ratio to some 15 or so decimal places in a running parlour game they had in their correspondence. Thus it was Cotes who developed the Naperian logarithms in his Logometria, referring them to Napirr as a honorific.

The constant ratio e was also known to De Moivre who made it the fundamental base of his logarithmic description of probability. De Moivre was the master at using the sine tables to solve incredible multinomialists equations, because he and Newton new a little secret: if you used the factorisation of unity then many terms cancel out, thus simplifying the equations, and putting them in terms of the sine or cosine values. It then became a task of locating these values in the sine tables to evaluate the multinomials. Newton used this trick to get De Moivre into the royal society, where he hoped to better his student and friends fortunes. Sadly he could not get him a stiprndiary seat, but it is a remarkable testimony to the devotion of De Moivre that he never left his friend all his life, and Newton supported him as best he could.

Cotes therefore was in the perfect place to make his ground breaking discovery, which he barely had time to share with De Moivre and Newton before dying unexpectedly. The observation was that the logarithm of the Cotes De Moivre theorem revealed a surprising result

i*arc x= ln{Cos(arc x) + isin(arc x)}

How he arrived at that through expansion of the logarithmic series I have not yet found out, but this was his identity, and it says that some constant conjoined with the arc length, and therefore some definite arc ratio i is the Naperian logarithm of the factors of unity.

Every form of calculus and analysis and measurement was harmoniously combined in that one iconic identity.

Today we hardly even realise it, using Euler' exponential formulation of this identity. But Cotes I am sure was more concerned with this form because it offered an alternative explanation of the orbits of objects around a central body. It was a quintessential gravitational identity which would revolutionise Newtonian gravitational theory, making calculations of orbits and ellipses etc that much easier. Besides, it identified the mysterious ratio i.

We have to realise how soon after its discovery he died, because De Moivre his close collaborator knew he believed the identity to be of fundamental significance but not how he meant to apply it. For me it gives the basic connection text Twistors and translational vectors, a point I will develop in another post.

So u^{}_{n} are labels for thes factors of unity, called the roots of unity because of the Cotes De Moivre theorem. The point is they are both factors and roots, thus harmonising factorisation and logarithms. The connection to factorisation means we can utilise polynomials, the connection to logarithms means we can use p-adic number systems, the nature of the factors means we can use vector notations, and these vectors are indistinguishable in space by supposed dimension of space, in fact they rightly determine the basis vectors of space as being of infinite dimension, and finally, because of the periodic clock arithmetic of the trig ratios these factors represent a modulo aritmetic Partitioning of space and quantity that uses them as basis vectors into modulo equivalence classes.

The modulo equivalence classes become an important orientation tool in this unfamiliar world of the children of Shunya.