Abstract
Quasicrystals are obtained as the projections of higher dimensional cubic lattices. This paper aims to show that the affine dihedral subgroup Wa(I2(h)) of the affine group Wa(Bn), h = 2n being the Coxeter number, offers a different perspective to -fold symmetric quasicrystallography. The affine group Wa(I2(h)) is constructed as the subgroup of the affine group Wa(Bn), the symmetry of the cubic lattice ℤn. The infinite discrete group with local dihedral symmetry of order 2 operates on the concentric -gons obtained by projecting the Voronoi cell of the cubic lattice with 2n vertices onto the Coxeter plane. Voronoi cells tiling the space lead to the tilings of the Coxeter plane with some overlaps of the rhombic tiles. After a general discussion on the lattice ℤn with its affine group Wa(Bn) embedding the affine dihedral group Wa(I2(h)) as a subgroup, its projection onto the Coxeter plane has been worked out with some examples. The cubic lattices with affine symmetry Wa(Bn), (n = 1, 2, 3, 4, 5) have been presented and shown that the projection of the lattice ℤ3 leads to the hexagonal lattice, the projection of the lattice ℤ4 describes the Ammann-Beenker quasicrystal lattice with 8-fold local symmetry and the projection of the lattice ℤ5 describes a quasicrystal structure with local 10-fold symmetry with thick and thin rhombi.