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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.                  From E8 to 4HD.                  4-dimensional HyperDiamond Lattice.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.


HyperDiamondLattices.

  n-dimensional HyperDiamond structures nHD areconstructed from Dn lattices. An n-dimensional HyperDiamond structures nHD is a latticeif and only if n is even. If n is odd, the nHD structure isonly a "packing", not a "lattice", because a nearest neighbor linkfrom an origin vertex to a destination vertexcannot be extended in the same directionto get anothernearest neighbor link. n-dimensional HyperDiamond structures nHD areconstructed from Dn lattices. The lattices of type Dn are n-dimensional Checkerboardlattices, that is,the alternate vertices of a Z^n hypercubic lattice.  

Conway and Sloane, in their book Sphere Packings, Lattices, andGroups (3rd edition, Springer, 1999).in chapter 4, section 7.3, pages119-120) define a packing

D+n = Dn u ( [1] + Dn )

[ where the gule vector [1] = (1/2, ... , 1/2) ]and say: "... D+n is a lattice packing if and only if n is even.

D+n is what David Finkelstein and I named a HyperDiamond latticenHD (although in odd dimensions it is technically only a packing andnot a lattice).

Conway and Sloane also say in chapter 4, section 7.1, page 117)that the lattice Dn is defined only for n greater than or equal to3.

To see what happens for n = 2, note that D2 should correspond tothe Lie algebra Spin(4),which is reducible to Spin(3)xSpin(3) =SU(2)xSU(2) = Sp(1)xSp(1) = S3xS3, and is not an irreducible Liealgebra. The root lattice of D2 is two copies of the root lattice ofSU(2), which is just a lattice of points uniformly distributed on aline.

If you are to fit the two lines together, you have to specify theangle at which they intersect each other, and requiring "latticestructure" or consistency with complex number multiplication does NOTunambiguously determine that angle: it can be either

60 degrees, which gives you the A2 root lattice
*--* / \ * * \ / *--*
of the Lie algebra SU(3) and the Eisenstein complex integers
or
90 degrees, which gives you the C2 = B2 root lattice
*--*--*| |* *| |*--*--*
of the Lie algebra Spin(5) = Sp(2) and the Gaussian complex integers.

Since a Dn lattice for n > 3 is a checkerboard, or halfof a hypercubic lattice, it is natural to define D2 as acheckerboard, or half of a C2 = B2 square lattice. Then the2-dimensional HyperDiamond lattice D+2 = D2 u ( [1] + D2 ) isseen to be the Z2 square lattice C2 = B2

where the orginal D2 is made up of the centers of the yellowsquares, the glue vectors are the (1/2,1/2) represented by pairs ofarrows, and the ( [1] + D2 ) is made up of the centers of thewhite squares.

The total 2-dim hyperdiamond structure is the Z2 integer lattice,sort of analogous to the 4-dim case in which D4 u ( [1] + D4) = Z4, so that

Note that the basic D2 structure is consistent with Feynman's2-dimensional checkerboard in which the lines of the checkerboardare 2-dim light-cone lines. 


Consider the 3-dimensional structure 3HD. Start with D3, the fcc close packing in 3-space. Make a second D3 shifted by the glue vector (0.5, 0.5, 0.5). Then form the union D3 u ([1] + D3). That is the 3-dimensional crystal structure that is made up of tetrahedral bonds:

The above figure (from Encyclopaedia Britannica CD-ROM 98, whichgot the figure from S.S. Zumdahl, Chemistry, 3rd ed., Heath (1993))is the crystal structure of H2O Water Ice (whose tetrahedral bondsfrom Oxygen to Oxygen are each half covalent and half Hydrogen) andof Carbon Diamonds (whose tetrahedral bonds from Carbon to Carbon arepurely covalent Carbon-Carbon).

HyperDiamond lattices were so named by David Finkelstein because of the 3-dimensional Diamond crystal structure.     

From E8 to 4HD.

 When you construct an 8-dimensional HyperDiamond 8HD lattice, you get D8 u  ([1] + D8) = E8.     The E8 lattice is in a sense fundamentally 4-dimensional, and the E8 HyperDiamond lattice naturally reduces to the 4-dimensional HyperDiamond lattice.   To get from E8 to 4HD, reduce each of the D8 lattices in the  E8 = 8HD = D8 u  ([1] + D8) lattice to D4 lattices.   We then get a 4-dimensional HyperDiamond              4HD = D4 u  ([1] + D4) lattice.   To see this, start with E8 = 8HD = D8 u  ([1] + D8).   We can write: D8 = { (  D4  ,0,0,0,0) u  (0,0,0,0,  D4  ) }                         u      {(1,0,0,0,1,0,0,0) +  (   D4  ,  D4  ) }    The third term is the diagonal term of an orthogonal decomposition of D8, and the first two terms are orthogonal to each other:  associative 4-dimensional Physical Spacetime and coassociative 4-dimensional Internal Symmetry space.  Now, we see that the orthogonal decomposition of 8-dimensional spacetime into 4-dimensional associative Physical Spacetime plus 4-dimensional Internal Symmetry space gives a decomposition of D8 into D4 + D4.    Since E8 = D8 u  ([1] + D8), and since [1] = (0.5,0.5,0.5,0.5,0.5,0.5,0.5,0.5) can be decomposed by [1] = (0.5,0.5,0.5,0.5,0,0,0,0) + (0,0,0,0,0.5,0.5,0.5,0.5) we have  E8 = D8 u  ([1] + D8})     =            (D4,0,0,0,0) + (0,0,0,0,D4))                              u (((0.5,0.5,0.5,0.5,0,0,0,0) + (0,0,0,0,0.5,0.5,0.5,0.5))                              +                ((D4,0,0,0,0) + (0,0,0,0,D4)))     = ((D4,0,0,0,0) u ((0.5,0.5,0.5,0.5,0,0,0,0) + (D4,0,0,0,0)))                              +     ((0,0,0,0,D4) u ((0,0,0,0,0.5,0.5,0.5,0.5) + (0,0,0,0,D4)))    Since 4HD is D4 u ([1] + D4),  E8 = 8HD = 4HDa + 4HDca where 4HDa is the 4-dimensional associative Physical Spacetime and 4HDca is the 4-dimensional coassociative Internal Symmetry space.   

4-dimensional HyperDiamondLattice.

 The 4-dimensional HyperDiamond lattice HyperDiamond is HyperDiamond = D4 u ([1] + D4). The 4-dimensional HyperDiamond HyperDiamond = D4 \cup ([1] + D4)is the Z^4 hypercubic lattice with null edges. It is the lattice that Michael Gibbs usedin his 1994 Georgia Tech Ph.D. thesis advised by David Finkelstein. The 8 nearest neighbors to the origin in the4-dimensional HyperDiamond HyperDiamond lattice can be writtenin octonion coordinates as: ( 1 + i + j + k) / 2 ( 1 + i - j - k) / 2 ( 1 - i + j - k) / 2 ( 1 - i - j + k) / 2 (- 1 - i + j + k) / 2 (- 1 + i - j + k) / 2 (- 1 + i + j - k) / 2 (- 1 - i - j - k) / 2   Here is an explicit construction of the 4-dimensionalHyperDiamond HyperDiamond lattice nearest neighbors to the origin.  Start with 24 vertices of a 24-CELL D4 with squared norm 2: +1   +1   0   0 +1   0   +1   0 +1   0   0   +1 +1   -1   0   0 +1   0   -1   0 +1   0   0   -1 -1   +1   0   0 -1   0   +1   0 -1   0   0   +1 -1   -1   0   0 -1   0   -1   0 -1   0   0   -1 0   +1   +1   0 0   +1   0   +1 0   +1   -1   0 0   +1   0   -1 0   -1   +1   0 0   -1   0   +1 0   -1   -1   0 0   -1   0   -1 0   0   +1   +1 0   0   +1   -1 0   0   -1   +1 0   0   -1   -1    Shift the 24 vertices by the glue vectorto get 24 more vertices [1] + D4: +1.5   +1.5   0.5   0.5 +1.5   0.5   +1.5   0.5 +1.5   0.5   0.5   +1.5 +1.5   -0.5   0.5   0.5 +1.5   0.5   -0.5   0.5 +1.5   0.5   0.5   -0.5 -0.5   +1.5   0.5   0.5 -0.5   0.5   +1.5   0.5 -0.5   0.5   0.5   +1.5 -0.5   -0.5   0.5   0.5 -0.5   0.5   -0.5   0.5 -0.5   0.5   0.5   -0.5 0.5   +1.5   +1.5   0.5 0.5   +1.5   0.5   +1.5 0.5   +1.5   -0.5   0.5 0.5   +1.5   0.5   -0.5 0.5   -0.5   +1.5   0.5 0.5   -0.5   0.5   +1.5 0.5   -0.5   -0.5   0.5 0.5   -0.5   0.5   -0.5 0.5   0.5   +1.5   +1.5 0.5   0.5   +1.5   -0.5 0.5   0.5   -0.5   +1.5 0.5   0.5   -0.5   -0.5    Of the 48 vertices of D4 u ([1] + D4),these 6 are nearest neighbors to the origin: -0.5   -0.5   0.5   0.5 -0.5   0.5   -0.5   0.5 -0.5   0.5   0.5   -0.5 0.5   -0.5   -0.5   0.5 0.5   -0.5   0.5   -0.5 0.5   0.5   -0.5   -0.5    Two more nearest neighbors, also of squared norm 1,of the origin   0.5    0.5    0.5    0.5 -0.5   -0.5   -0.5   -0.5    come from adding the glue vector to the origin 0   0   0   0   and to the squared norm 4 point  -1   -1   -1   -1  Since the D4-D5-E6-E7-E8 VoDou Physics model is fundamentally a Planck Scale HyperDiamond Lattice Generalized Feynman Checkerboard model, it violates Lorentz Invariance at the Planck Scale, affecting Ultra High Energy Cosmic Rays. 

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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