Affine Dihedral Subgroups of Higher Dimensional Cubic Lattices Zn\mathbb{Z}^nZn and Quasicrystallography (original) (raw)

Quasicrystals described as the projections of higher dimensional cubic lattices, and the particular affine extension (2 (ℎ)) of the dihedral group (2 (ℎ)) of order 2h, ℎ = 2 being the Coxeter number, as a subgroup of affine group () offers a different perspective to h-fold symmetric quasicrystallography. The affine group (2 (ℎ)) is constructed as the subgroup of the affine group (), the symmetry of the cubic lattice ℤ. The infinite discrete group with local dihedral symmetry of order 2h operates on the concentric h-gons obtained by projecting the Voronoi cell of the cubic lattice with 2 vertices onto the Coxeter plane. Voronoi cells tile the space facet to facet, consequently, leading to the tilings of the Coxeter plane with some overlaps of the rhombic tiles. It is noted that the projected Voronoi cell is the overlap of h copies of the h-gons tiled with some rhombi and rotated by the angle 2 ℎ. After a general discussion on the lattice ℤ with its affine group () embedding the affine dihedral group (2 (ℎ)) as a subgroup, its projection onto the Coxeter plane has been worked out with some examples. The cubic lattices with affine symmetry (), (= 1,2,3,4,5) have been presented and shown that the projection of the lattice ℤ 3 leads to the hexagonal lattice, the projection of the lattice ℤ 4 describes the Amman-Beenker quasicrystal lattice with 8-fold local symmetry and the projection of the lattice ℤ 5 describes a quasicrystal structure with local 10fold symmetry with thick and thin rhombi. It is then straight forward to show that the projections of the cubic lattices with even higher dimensions onto the Coxeter plane may lead to the quasicrystal structures with 12-fold, 18-fold symmetries and so on.