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Clifford Tensor Product Universe

 

The beginning of our physical universemay be described by a D4-D5-E6-E7-E8 VoDouPhysics model generalization ( related to loopoids) of the von Neumann hyperfiniteII1 Clifford tensor product

... x Cl x Cl x Cl xCl x Cl x Cl x Cl x ...

where Cl = Mat2(C) to a similar structure with Cl = Mat16(R).

... Cl--Cl--Cl--Cl--Cl--Cl--Cl ...
that corresponds to the 1-dim lattice of Natural Numbers.

Note that in the case of the example show abovewhere each vertex neighborhood + looks in the continuum limit like aunit disk of the complex plane, all the vertex neighborhoods are thesame, and the total 2-dim lattice space looks in the continuum limitlike a big complex plane with its unique differential structure,

while in the case of each vertex neighborhood +looking like an E8lattice (in the continuum limit like the8-dimensional vector space of the Cl(8,R) = Mat16(R)) there can be 7different kinds of vertex neighborhoods, corresponding to the7 differentE8 latttices. Further, if in the continuumlimit the boundary of each vertex neighborhood looks like a 7-sphereS7, then, since each S7can have 28 different differentialstructures, the total 8-dim space can havea very complicated structure, whether viewed as a lattice (withvarying types of E8 neighborhoods) or in the continuum limit (asa manifoldwith complicated Riemannian structure).

It is interesting that the 2-dimensionalweave structure looks a lot like a Ulam spiral.

According to anAbarim web page:

"... Stanislaw Ulam was attending some boring meeting, and to divert himself somewhat he began to scribble on a piece of paper. ... He put down the number 1 as the bright shining center of a universe of numbers that Big Banged outwardly in a spiral:

Much to his amazement the prime numbers appeared to gravitate towards diagonal lines emanating from the central 1. ... Most of them sat on or in the vicinity of a diagonal, but some obviously didn't. ...".

According to aPrime Number Spiral web page:

"... Consider a rectangular grid. We start with the central point and arrange the positive integers in a spiral fashion (anticlockwise) as at right. The prime numbers are then marked ... There is a tendency for the prime numbers to form diagonal lines. This can be seen more clearly in the image below,

which shows a window onto a square array of 640 x 640 numbers, with the primes marked by white pixels. ...".

According to aM. Watkins web page:

"... There is currently no explanation for the distinct diagonal lines which appear when the primes are marked out along a particular 'square spiral' path. ...".

Physically, if the 2-dim weave corresponds to 2-dim spacetime, thediagonal lines would correspond to light-cone correlationsof points of spacetime.

In dimensions greater than 2, theweaving should produce something like a Moore space-fillingcurve. Acccording to aweb page of V. B. Balayoghan:

"... The Hilbert and Moore curves use square cells -- the level n curve has 4^n cells (and hence 4^n - 1 lines). The Moore curve has the same recursive structure as the Hilbert curve, but ends one cell away from where it started. The Hilbert curve starts and ends at opposite ends of a side of the unit square. ...".

According to aweb page by William Gilbert:

"... We exhibit a direct generalization of Hilbert's curve that fills a cube. The first three iterates of this curve are shown.

In constructing one iterate from the previous one, note that the direction of the curve determines the orientation of the smaller cubes inside the larger one.

The initial stage of this three dimensional curve can be considered as coming from the 3-bit reflected Gray code which traverses the 3-digit binary strings in such a way that each string differs from its predecessor in a single position by the addition or subtraction of 1. The kth iterate could be considered a a generalized Gray code on the Cartesian product set {0,1,2,...,2^k-1}^3.

The n-bit reflected binary Gray code will describe a path on the edges of an n-dimensional cube that can be used as the initial stage of a Hilbert curve that will fill an n-dimensional cube. ...".

According to Numerical Recipes in C, by Press, Teukolsky,Vettering, and Flannery (2nd ed, Cambridge 1992):

"... A Gray code is a function G(i) of the integers i, that for each integer N > 0 is one-to-one for 0 < i < 2^N -1, and that has the following remarkable property: The binary representation of G(i)and G(i+1) differ in exactly one bit. an example of a Gray code ... is the sequence ...[
0000 ( 0=0000), 0001 ( 1=0001), 0011 ( 2=0010), 0010 ( 3=0011),   0110 ( 4=0100), 0111 ( 5=0101), 0101 ( 6=0110), 0100 ( 7=0111),   1100 ( 8=1000), 1101 ( 9=1001), 1111 (10=1010), 1110 (11=1011),   1010 (12=1100), 1011 (13=1101), 1001 (14=1110), 1000 (15=1111) 

]... for i = 0, ... 15. The algorithm for generating this code is simply to form ... XOR of i with 1/2 (integer part). ... G(i) and G(1+1) differ in the bit position of the rightmosst zero bit of i ... Gray codes can be useful when you need to do some task that depends intimatelyu on the bits of i, looping over many values of i. Then, if there are economies in repeating the task for values differing only by one bit, it makes sense to do things in Gray code order rather than consecutive order. ...".

According to some MathWorldweb pages:

"... The binary reflected Gray code is closely related to the solutions of the towers of Hanoi and baguenaudier, as well as to Hamiltonian circuits of hypercube graphs ...[ A Hamiltonian Circuit is]... A graph cycle (i.e., closed loop) through a graph that visits each node exactly once ... The number of Hamiltonian circuits on an n-hypercube is 2, 8, 96, 43008, ...".

 

If you look at a 2-dimensional slice of the n-dimensional Moorecurve including the time axis and one spatial axis, you see somethinglike a Ulam Spiral and also like a2-dimensional Feynman checkerboard.

 

The Ising model( shown by Hal Gersch (Int. J. Theor. Phys. 20(1981) 491) to be equivalent to aFeynman checkerboard for the 2-dim Diracequation ) is relatedto von Neumann algebras:

 


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