الملخص الإنجليزي
Quasicrystallography can be described as the projections of higher dimensional cubic lattices, where the two-dimensional quasicrystallography presents the local ℎ-fold symmetry described by rhombic tiles. Affine dihedral group 𝑊𝑎(𝐼2(ℎ)) of order 2h, is constructed as a subgroup of affine 𝑊(𝐵𝑛) that gives a different perspective to ℎ-fold symmetric quasicrystallography. The Voronoi cell of the cubic lattice ℤ𝑛 (𝑛 = 1, 2, 3, 4, 5, 6) with 2𝑛 vertices is projected onto the Coxeter plane. The projection of the lattice ℤ𝑛 onto the Coxeter plane is made by the affine dihedral group 𝑊(𝐼2(ℎ)). The projection of ℤ5 displays a quasicrystal structure with 10-fold symmetric tiling, the projection of ℤ6 displays a quasicrystal structure with 12-fold symmetric tiling. Affine dihedral subgroup 𝑊(𝐼2(ℎ)) plays an important role in the tiling system.
