Aperiodic Tilings in 2, 3, and 4dimensions can be thought of as Irrational Slices of an 8-dimensionalE8 Lattice and its sublattices, such as E6.The 2-dimensional Penrose Tiling in theabove image was generated by Quasitiler asa section of a 5-dimensional cubic lattice based on the 5-dimensionalHyperCube shown in the center above thePenrose Tiling plane. The above-plane geometric structures in theabove image are, going from left to right:
The 240 E8 root vectors form a WittingPolytope. They are related to the 256 elements of the Cl(1,7)Clifford Algebra of theD4-D5-E6-E7-E8 VoDou Physics model as follows:
- 4-dimensional 24-cell, whose 24 vertices are the root vectors of the 24+4 = 28-dimensional D4 Lie algebra;
- two 4-dimensional HyperOctahedra, lying (in a 5th dimension) above and below the 24-cell, whose 8+8 = 16 vertices add to the 24 D4 root vectors to make up the 40 root vectors of the 40+5 = 45-dimensional D5 Lie algebra;
- 5-dimensional HyperCube, half of whose 32 vertices are lying (in a 6th dimension) above and half below the 40 D5 root vectors, whose 16+16 = 32 vertices add to the 40 D5 root vectors to make up the 72 root vectors of the 72+6 = 78-dimensional E6 Lie algebra;
- two 27-dimensional 6-dimensional figures, lying (in a 7th dimension) above and below the the 72 E6 root vectors, whose 27+27 = 54 vertices add to the 72 E6 root vectors to make up the 126 root vectors of the 126+7 = 133-dimensional E7 Lie algebra; and
- two 56-dimensional 7-dimensional figures, lying (in an 8th dimension) above and below the the 126 E7 root vectors, and two polar points also lying above and below the 126 E7 root vectors, whose56+56+1+1 = 114 vertices add to the 126 E7 root vectors to make up the 240 root vectors of the 240+8 = 248-dimensional E8 Lie algebra.
1 8 28 56 70 56 28 8 1
- Cl(1,7) has 256 = 2^8 elements, corresponding to the 2^8 vertices of an 8-dimensional HyperCube and having the graded structure
with even part 1 28 70 28 1 and odd part 8 56 56 8
- An 8-dimensional HyperCube decomposes into 2 half-HyperCubes, each with 128 vertices, and each corresponding to one of the 2 mirror-image half-spinor representations of the D8 Lie algebra whose Euclidean-signature spin group is 120-dimensional Spin(16) and whose half-spinor representations have dimension (1/2)(2^(16/2)) = 256/2 = 128;
- One half-HyperCube corresponds to the 128 even elements 1 28 70 28 1 and the other to the 128 odd elements 8 56 56 8;
- The128 vertices of the odd 8-dimensional half-HyperCube with graded structure 8 56 56 8 correspond to 128 of the 240 E8 root vectors as follows:
- 8 to the 8 octonion vector space basis elements, with positive sign;
- 8 to the 8 octonion vector space basis elements, with negative sign;
- 56+56 = 112 to the 112 vectors with non-zero components on the octonion real axis;
- The other 240-128 = 112 do not directly correspond to the 128 vertices of the even 8-dimensional HyperCube of the even half-spinor representation of the D8 Spin(16) Lie algebra, but correspond to 112 of the 120 generators of Spin(16), the adjoint bivector representation of D8.
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