Karel Culik II has presented An Aperiodic Set of 13 Wang Tiles.
Consider the 6-dim subspace spanned by {i,j,k,I,J,K}.
The 3-dim {i,j,k} subspace is the spatial part of
the 4-dim rational {1,i,j,k} quaternionic spacetime.
It is the associative 4-dim spacetime of the
D4-D5-E6 model.
The 3-dim {I,J,K} subspace is the spatial part of
the 4-dim irrational Golden {E,I,J,K} quaternionic spacetime.
It is the coassociative 4-dim internal symmetry space of the
D4-D5-E6 model.
The 6-dim {i,j,k,I,J,K} space has 3-dim subspaces
that can correspond to our 3-dim physical space,
and we can look at the section of a 6-dim hypercubic lattice
that is in such 3-dim subspaces.
One such subspace is the 3-dim space {i,j,k},
for which the 3-dim section of the 6-dim hypercubic lattice
is a 3-dim cubic lattice.
Another such 3-dim subspace gives
a 3-dim face-centered-cubic fcc lattice.
Still another 3-dim subspace gives
for each 6-dim hypercube
a rhombic triacontahedron
whose 30 rhombic faces each have Golden diagonal ratio PHI:1 A rhombic triacontahedron is the dual of the icosidodecahedron and can be made by truncating the 30 edges of an icosahedron. Therefore a rhombic triacontahedron is intermediate between a cube of a cubic lattice and a truncated octahedron associated with part of an fcc lattice just as an icosahedron is intermediate between an octahedron and a cuboctahedron with respect to Fuller jitterbug or tensegrity transformations. Lalvani at NYIT (see Connections, by Kappraff) has shown
a continuous transformation based on 3-dim sections of a 6-dim hypercube from cube to rhombic triacontahedron to truncated octahedron. Although rhombic triacontahedra do not fill space, just as truncated icosahedra do not fill space, 3-dim space can be filled by left-handed and right-handed fists with Golden rhombus faces:
The tiling of 3-dim space by left and right fists
projects onto 2-dim space to produce an aperiodic Penrose tiling. The program QuasiTiler produces Penrose tilings by taking 2-dim slices of a 5-dim hypercubic lattice, rather than the 2-step process of 6-dim to 3-dim to 2-dim described above, but the results are the same. Penrose tilings are made up of kites and darts:
Question: There are 7 different kinds of vertex neighborhoods of Penrose tilings by kites and darts.
Are they related to the 7 imaginary octonions? In their book, Tilings and Patterns, Grunbaum and Shephard show that a tiling by Penrose kites and darts can be cut up into 8 types of tiles:
These 8 types of tiles have 28 different kinds of edges:
There are 16 vertical edges and 12 horizontal edges. Questions: Do the 8 types of tiles correspond to the 8-dim vector and half-spinor representations of Spin(0,8)? Do the 28 edges correspond to the 28-dim adjoint representation of Spin(0,8)? Do the 16 vertical edges correspond to the 16-dim U(4) = Spin(0,6) x U(1) that gives conformal gravity by the MacDowell-Mansouri mechanism in the D4-D5-E6 model? Do the 12 horizontal edges correspond to the 12-dim Standard Model SU(3)xSU(2)xU(1) of the D4-D5-E6 model?