The fact that the factors of unity are also the roots of unity harmonises many apparently disparate calculii, this Roger Cotes was assured of in his own mind.

The periodicity of the trig ratios in the unit circle enable the modulo arithmetics to be encribed within the circumference, there is in fact a one to one mapping between any set of ordered points finite or infinite that can be encribed in the unit circles periphery, theeir is a one to one mapping of all points in space to the point on or in or involved with the unit sphere; and one can attribute a hyperbolic geometry to sets of points within and without the unit sphere pr a hupobolic geometry of the trig ratios entirely within the unit sphere. Finally we can attribute the basis of all vector spaces to the polynomial factors of unity, that is the roots of unity with there intense modulo arithmetics. Thus the quaternions of Hamilton form the first directly derived basis of a vector space from the unit sphere. It is not the first reference frame based on the sphere, because that geometry was worked out millennia ago in Persia where spherical geometry and spherical trigonometry reached its zenith. But the Grassmanian gauss conception of a manifold was definitely not the basis of these Persian geometrists, or was it? We cannot readily dismiss there concepts as being radically different from our own, despite the surface language differences.

In any case, it was Hamilton who first successfully revealed the structure of the roots of unity within the unit sphere. Grassmann's more general approach actually did not include the imaginsries formally until his 1862 rewrite, although he made a tantalising foray into Euler' s exponential partition of geometries in space which he never came good on. Hamilton's quaternions are the roots of unity applied in their spherical form, and the difference is profound.

I am doing this analysis to check if Hamilton got it all right, because the children of Shunya are very numerous and agile!

The children of Shunya are defined in every orient able disk plane, but they dwell in every cylindrical space. And the union of every cylindrical space is the spherical space of the unit sphere. The cone of conical curves all exist there and the direct link to every curved surface resides in that space, and the children of Shunya can describe infinite possibilities. Which possibility we experience is determined by the constraints we establish.

In our world people have been killed one supposes, to establish the veracity of the Pythagoras theorem. Thus we do not accept constraints where the magnitude^{2} of the combination of the unit vectors is different from n*1 . This corresponds to 2*the sum of the logarithms of the base terms being = 0 mod the root type. This ensures that the coefficients can represent orthogonal coordinates in 3d space by maintaining a conformal mapping under the underlying algebra. Then one also has to specify constraints on the coefficients so they behave in the same way as a Cartesian coordinate system. Or if preferred they can be made to conform to a polar coordinate system. So we have two main constraints on the system Lagrangian for it to model 3d space. Typically the number of dimensions of the basis vector is not the determining factor. The coefficient behaviour is the determining restraint. Thus if the scalar field ie the coefficient field, has some specified constraint that is not consistent with 3d space that constraint will be the cause of odd behaviour, not the dimensional count.

The constraint on the factors of unity typically mean that the unit sphere is not the source of n dimensional vector bases. They are generated from spheres within the unit sphere. Thus an infinite vector basis ld be equivalent to a point generating the basis. Instead of using that image we tend to keep the unit sphere and decrease the coefficient scale factor so the scalar is a "point" scale. The effect of this is to generate minuscule transformations on the surface of the unit sphere. And these are labelled as vector dimensions. Thus an n dimensional vector basis is really a spherical surface with n equally spaced nodes on it representing the radial orientations from the centre. However, if we restrict this machinery to the circle we have n points equally spaced around the periphery . How do we know which scenario we are in?

Hamilton was the first person to solve that problem, although grassmann later wrote down the full theoretical method from the viewpoint of a consistent analytical treatment. Hamilton did not exactly have a lucky guess, but he was lucky nonetheless to be working in a space wit axes that partially obscured the true size of the problem. It was only after he had his insight that he realised the connection to De Moivre's work, and of course he knew very little of Cotes. Consequently he glimpsed the general connections but had no inclination to progress much further for his own personal reasons. He devoted himself to quaternions and poetry and religion. Grassmann on the other hand had much to do to make his way in the world, and fully hoped his thorough treatment of the manifold concept he had developed would be his ticket to advancement. Unfortunately Gauss felt otherwise. Nevertheless, Grassmann's firm conviction in the ideas of his father, sustained him through a very difficult period of his life. His fathers ideas on dynamic space, not merely his fathers in isolation I might add, but those of an innovative group of thinkers about mechanical dynamsms in space, especially in crystallography, we're well developed enough to guide Herrman over considerable technical difficulties, and to establish a robust framework for his developing ideas. However Gauss did not publish his ideas on imaginaries until 1830. After his father had set out his theory on the differential structure of Zahlen.mby the time Hermann had gotten around to reading gauss ideas, Hermann had already published Der Ausdehnungslehre 1844. Gauss's choice of the linear combinatorial form for the general imaginary quantity only served to confirm herrmann's conception and analysis. However, the expected call from Gauss never came. In fact only a fell and dishevelled rebuff was received! Herrmann was nearly a broken man by the time Riemann came to visit him, not to give him encouragement, but to insist he write he material in the standard format.mthis was a backhanded compliment, I suppose, but Herrmann had little choice but to accept his "deal" . In 1862 he cries out for vindication by later history, but in fact he gets a little justice after rimannian dies. His work is more general and speculative in that it posits general magnitudes with general orientations. There s no evidence that he tied it to the work of De Moivre and Cotes as Hamilton did, in which case his general approach is an alternative description of the concept of orientation in space, and how it might work algebraically..

When I discuss the mechanics of the De Moivre Cotes method it sounds like group theory. This is because both Hamilton and Eulrr established the framework language for group theory: Hamilton's algebraic mathesis of the imaginaries, and Euler's modulo arithmetic of the cyclic groups of the integers.

When I consider the nth roots of unity I in fact consider the cyclic group of the integers modulo n. thus modulo arithmetic is involved from the get go.

u^{i}_{n} are the n nth roots of unity or the n factors of unity. This is the modulo n cyclic group of factors of unity and I can write

Log_{un}{u^{i}_{n}} = i mod n

This is just to emphasise that the factors of unity are a mod n cyclic group of the integers which can be written exponentially or logarithmically .

To establish these as a basis vector I have to check they are independent of each other, which as a polynomial in the nth root they can be shown to be. I then have to linearly combine them with some coefficients a^{}_{i}, and then set out the rules or constraints for these combinations.

To make the polynomial basis a unit vector in space of n dimensions I define modulus {U}^{2}

Where U = P(u^{}_{n}) as Sum{a^{2}_{i}} and divide U by mod{U}

The constraint on the sum of 2 times the logarithm sum of the roots of unity is to ensure we have a unit sphere/circle, and the constraint on the modulus protects Pythagorean principles. What it also does is cause space to expand in a Theodorus spiral, the more dimensions we use. The spiral however is disposed in 3d and therefore behaves differently in real space than it does on the plane.

The spiral is an important indicator of what space we are in. If the spiral remains strictly in the plane we note increasing expansion and contraction. However if the spiral has more degrees of freedom we should notice a range of complex behaviours Brownian motion being one, vortices being another and every behaviour in between.

Now I have to decide if I want contra vectors, in which case I can simplify by choosing roots of unity with cyclic groups mod 2 n, and then I choose those factors which have a modulo n relationship as pairing candidates. While contra relationships are normal to our axes systems, they are not necessary to describe position or motion in space. Both ted and Kujonai developed rules for dealing with the non standard axial systems which ted called polysigns.

Once all this machinery is set up, including checks for associativity and commutativity, and divisibility one should have a serviceable reference frame system which one attaches to a known 3d system to ensure calibration.

There are only 5 regular solid nets to choose from according to the Platonic solids, and current modern maths. This effectively means we have only 5 reference frames that we can sensibly build from the roots of unity which will operate in real space. Thus we have another clue, besides spirals, and that is the shapes of the platonic solids.

Whatever vector space description of our reality we wish to build they must reduce to these 5 distinct solids. Thus Keplers harmony of the spheres will prove to fit the description of motions in space, simply because he used the only 5 possible reference frames to construct it. It is to be noted that not all of them have contra axes and some are better for rotations than others.

Now of course we probably have solutions based on imaginary magnitudes, but thes will not be evenly distributed in space, but evenly distributed relative to the surface of the sphere.mthus they will tend to have exponential or logarithmic behaviours when mapped into 3d space as we like it!