Aperiodic Tilings in 2, 3, and 4
dimensions can be thought of as Irrational Slices of an 8-dimensional
E8 Lattice and its sublattices, such as E6.
The 2-dimensional Penrose Tiling in the
above image was generated by Quasitiler as
a section of a 5-dimensional cubic lattice based on the 5-dimensional
HyperCube shown in the center above the
Penrose Tiling plane. The above-plane geometric structures in the
above image are, going from left to right:
- 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.
The 240 E8 root vectors form a Witting
Polytope. They are related to the 256 elements of the Cl(1,7)
Clifford Algebra of the
D4-D5-E6-E7-E8 VoDou Physics model as follows:
- Cl(1,7) has 256 = 2^8 elements, corresponding to the 2^8
vertices of an 8-dimensional HyperCube and
having the graded structure
1 8 28 56 70 56 28 8 1
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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