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From Sets toQuarks:

Deriving the Standard Model plus Gravitation from Simple Operations on Finite Setsby Tony Smith Table of Contents: Chapter 1 - Introduction. Chapter 2 - From Sets to Clifford Algebras.                   MANY-WORLDS QUANTUM THEORY.   Chapter 3 - Octonions and E8 lattices. Chapter 4 - E8 spacetime and particles. Chapter 5 - HyperDiamond Lattices. Chapter 6 - Spacetime and Internal Symmetry Space. Chapter 7 - Feynman Checkerboards. Chapter 8 - Charge = Amplitude to Emit Gauge Boson. Chapter 9 - Mass = Amplitude to Change Direction. Chapter 10 - Protons, Pions, and Physical Gravitons. Appendix A - Errata for Earlier Papers. References.


Octonions and E8 lattices.

 The E8 lattice is made up of onehypercubic Checkerboard D8 lattice plusanother D8 shifted by a glue vector(1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2),  so that E8 = D8 u  ([1] + D8) is an 8-dimensional HyperDiamond 8HD lattice.   If octonionic coordinate are chosen so thata given minimal vector in E8 is +1,the vectors in E8 that are perpendicular to +1 make upa spacelike E7 lattice.  The E8 lattice nearest neighbor vertices haveonly 4 non-zero coordinates,like 4-dimensional spacetime with speed of lightc = sqrt3, rather than 8 non-zero coordinates,like 8-dimensional spacetime with speed of light c = sqrt7,so the E8 lattice light-cone structure appears to be4-dimensional rather than 8-dimensional. To build the E8 Lattice: Begin with an 8-dimensional octonionic spacetime R^8,where a basis for the octonions is { 1,i,j,k,E,I,J,K } . The vertices of the E8 lattice are of the form (a01 + a1E + a2i + a3j + a4I + a5K + a6k + a7J)/2 , where the ai may be either all even integers, all odd integers, or four of each (even and odd), with residues mod 2 in the four-integer cases being (1;0,0,0,1,1,0,1) or (0;1,1,1,0,0,1,0) or the same with the last seven cyclically permuted.E8 forms an integral domain of integral octonions. The E8 lattice integral domain has 240 units:  +/- 1, +/- i, +/- j, +/- k +/- E, +/- I, +/- J +/- K, ( +/- 1 +/- I +/- J +/- K)/2, ( +/- E +/- i +/- j +/- k)/2, and the last two with cyclical permutations of { i,j,k,E,I,J,K } in the order (E, i, j, I, K, k, J). The cyclical permutation (E, i, j, I, K, k, J)preserves the integral domain E8,but is not an automorphism of the octonionssince it takes the associative triad { i,j,k }into the anti-associative triad { j,ie,je }. The cyclical permutation (E, I, J, i, k, K, j)is an automorphism of the octonions buttakes the E8 integral domain defined aboveinto another of seven integral domains. Denote the integral domain described above as 7E8, and the other six by iE8 , i = 1, ... , 6.  The 240 units of the 7E8 lattice corresponding tothe integral domain 7E8 represent the 240 lattice pointsin the shell at unit distance (also commonly normalized as 2):  +/- 1, +/- i, +/- j, +/- k, +/- E, +/- I, +/- J, +/- K,(+/- 1 +/- I +/- J +/- K)/2(+/- e +/- i +/- j +/- k)/2(+/- 1 +/- K +/- E +/- k)/2(+/- i +/- j +/- I +/- J)/2(+/- 1 +/- k +/- i +/- J)/2(+/- j +/- I +/- K +/- E)/2(+/- 1 +/- J +/- j +/- E)/2(+/- I +/- K +/- k +/- i)/2(+/- 1 +/- E +/- I +/- i)/2(+/- K +/- k +/- J +/- j)/2(+/- 1 +/- i +/- K +/- j)/2(+/- k +/- J +/- E +/- I)/2(+/- 1 +/- j +/- k +/- I)/2(+/- J +/- E +/- i +/- K)/2    The other six integral domains iE8 are:   1E8:  +/- 1, +/- i, +/- j, +/- k, +/- E, +/- I, +/- J, +/- K, (+/- 1 +/- J +/- i +/- j)/2(+/- k +/- E +/- I +/- K)/2(+/- 1 +/- j +/- I +/- K)/2(+/- i +/- k +/- E +/- J)/2(+/- 1 +/- K +/- k +/- i)/2(+/- j +/- E +/- I +/- J)/2(+/- 1 +/- i +/- E +/- I)/2(+/- j +/- k +/- J +/- K)/2(+/- 1 +/- I +/- J +/- k)/2(+/- i +/- j +/- E +/- K)/2(+/- 1 +/- k +/- j +/- E)/2(+/- i +/- I +/- J +/- K)/2(+/- 1 +/- E +/- K +/- J)/2(+/- i +/- j +/- k +/- I)/2     2E8:  +/- 1, +/- i, +/- j, +/- k, +/- E, +/- I, +/- J, +/- K,(+/- 1 +/- i +/- k +/- E)/2(+/- j +/- I +/- J +/- K)/2(+/- 1 +/- E +/- J +/- j)/2(+/- i +/- k +/- I +/- K)/2(+/- 1 +/- j +/- K +/- k)/2(+/- i +/- E +/- I +/- J)/2(+/- 1 +/- k +/- I +/- J)/2(+/- i +/- j +/- E +/- I)/2(+/- 1 +/- J +/- i +/- K)/2(+/- j +/- k +/- E +/- I)/2(+/- 1 +/- K +/- E +/- I)/2(+/- i +/- j +/- k +/- J)/2(+/- 1 +/- I +/- j +/- i)/2(+/- k +/- E +/- J +/- K)/2     3E8:  +/- 1, +/- i, +/- j, +/- k, +/- E, +/- I, +/- J, +/- K,(+/- 1 +/- k +/- K +/- I)/2(+/- i +/- j +/- E +/- J)/2(+/- 1 +/- I +/- i +/- E)/2(+/- j +/- k +/- J +/- K)/2(+/- 1 +/- E +/- j +/- K)/2(+/- i +/- k +/- I +/- J)/2(+/- 1 +/- K +/- J +/- i)/2(+/- j +/- k +/- E +/- I)/2(+/- 1 +/- i +/- k +/- j)/2(+/- e +/- I +/- J +/- K)/2(+/- 1 +/- j +/- I +/- J)/2(+/- i +/- k +/- E +/- K)/2(+/- 1 +/- J +/- E +/- k)/2(+/- i +/- j +/- I +/- K)/2     4E8:  +/- 1, +/- i, +/- j, +/- k, +/- E, +/- I, +/- J, +/- K,(+/- 1 +/- K +/- j +/- J)/2(+/- i +/- k +/- E +/- I)/2(+/- 1 +/- J +/- k +/- I)/2(+/- i +/- j +/- E +/- K)/2(+/- 1 +/- I +/- E +/- j)/2(+/- i +/- k +/- J +/- K)/2(+/- 1 +/- j +/- i +/- k)/2(+/- e +/- I +/- J +/- K)/2(+/- 1 +/- k +/- K +/- E)/2(+/- i +/- j +/- I +/- J)/2(+/- 1 +/- E +/- J +/- i)/2(+/- j +/- k +/- I +/- K)/2(+/- 1 +/- i +/- I +/- K)/2(+/- j +/- k +/- E +/- J)/2     5E8:  +/- 1, +/- i, +/- j, +/- k, +/- E, +/- I, +/- J, +/- K,(+/- 1 +/- j +/- E +/- i)/2(+/- k +/- I +/- J +/- K)/2(+/- 1 +/- i +/- K +/- J)/2(+/- j +/- k +/- E +/- I)/2(+/- 1 +/- J +/- I +/- E)/2(+/- i +/- j +/- k +/- K)/2(+/- 1 +/- E +/- k +/- K)/2(+/- i +/- j +/- I +/- J)/2(+/- 1 +/- K +/- j +/- I)/2(+/- i +/- k +/- E +/- J)/2(+/- 1 +/- I +/- i +/- k)/2(+/- j +/- E +/- J +/- K)/2(+/- 1 +/- k +/- J +/- j)/2(+/- i +/- E +/- I +/- K)/2     6E8:  +/- 1, +/- i, +/- j, +/- k, +/- E, +/- I, +/- J, +/- K,(+/- 1 +/- E +/- I +/- k)/2(+/- i +/- j +/- J +/- K)/2(+/- 1 +/- k +/- j +/- i)/2(+/- e +/- I +/- J +/- K)/2(+/- 1 +/- i +/- J +/- I)/2(+/- j +/- k +/- E +/- K)/2(+/- 1 +/- I +/- K +/- j)/2(+/- i +/- k +/- E +/- J)/2(+/- 1 +/- j +/- E +/- J)/2(+/- i +/- k +/- I +/- K)/2(+/- 1 +/- J +/- k +/- K)/2(+/- i +/- j +/- E +/- I)/2(+/- 1 +/- K +/- i +/- E)/2(+/- j +/- k +/- I +/- J)/2      The vertices that appear in more than one lattice are: +/- 1, +/- i, +/- j, +/- k, +/- E, +/- I, +/- J, +/- K in all of them; (+/- 1 +/- i +/- j +/- k)/2 and (+/- e +/- I +/- J +/- K)/2 in 3E8, 4E8, and 6E8; (+/- 1 +/- i +/- E +/- I)/2 and (+/- j +/- k +/- J +/- K)/2 in 7E8, 1E8, and 3E8; (+/- 1 +/- j +/- E +/- J)/2 and (+/- i +/- k+/- I +/- K)/2 in 7E8, 2E8, and 6E8; (+/- 1 +/- k +/- E +/- K)/2 and (+/- i +/- j +/- I +/- J)/2 in 7E8, 4E8, and 5E8; (+/- 1 +/- i +/- J +/- K)/2 and (+/- j +/- k +/- E +/- I)/2 in 2E8, 3E8, and 5E8; (+/- 1 +/- j +/- I +/- K)/2 and (+/- i +/- k+/- e +/- J)/2 in 1E8, 5E8, and 6E8; (+/- 1 +/- k +/- I +/- J)/2 and (+/- i +/- j +/- E +/- K)/2 in 1E8, 2E8, and 4E8;  The 240 unit vertices in the E8 lattices do not includeany of the 256 E8 light cone vertices,of the form (+/- 1 +/- i +/- j +/- k +/- E +/- I +/- J +/- K)/2. They appear in the next layer out from the origin,at radius sqrt 2, which layer contains in all 2160 vertices.  
The E8 lattice is, in a sense, fundamentally 4-dimensional.   For instance:    the E8 lattice nearest neighbor vertices have only 4 non-zero coordinates,like 4-dimensional spacetime with speed of light c = sqrt(3),rather than 8 non-zero coordinates,like 8-dimensional spacetime with speed of light c = sqrt(7),so the E8 lattice light-cone structure appears to be4-dimensional rather than 8-dimensional;   the representation of the E8 lattice by quaternionic icosians, as described by Conway and Sloane;   the Golden ratio construction of the E8 lattice fromthe D4 lattice, which has a 24-cell nearest neighbor polytopeThe construction starts with the 24 vertices of a 24-cell,then adds Golden ratio points on each of the 96 edges of the 24-cell, then extends the space to 8 dimensions by considering the algebraicaly independent sqrt(5) part of the coordinates to be geometrically independent, andfinally doubling the resulting 120 vertices in 8-dimensionalspace by considering both the D4 lattice and its dual D4* to get the 240 vertices of the E8 lattice nearest neighborpolytope (the Witting polytope); and   the fact that the 240-vertex Witting polytope,the E8 lattice nearest neighbor polytope,most naturally lives in 4 complex dimensions,where it is self-dual, rather than in 8 real dimensions.    
From Sets to Quarks: Table of Contents: Chapter 1 - Introduction. Chapter 2 - From Sets to Clifford Algebras.   MANY-WORLDS QUANTUM THEORY.  Chapter 3 - Octonions and E8 lattices. Chapter 4 - E8 spacetime and particles. Chapter 5 - HyperDiamond Lattices. Chapter 6 - Spacetime and Internal Symmetry Space. Chapter 7 - Feynman Checkerboards. Chapter 8 - Charge = Amplitude to Emit Gauge Boson. Chapter 9 - Mass = Amplitude to Change Direction. Chapter 10 - Protons, Pions, and Physical Gravitons. Appendix A - Errata for Earlier Papers. References.
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