From Affine 𝑨𝟒 to Affine 𝑯𝟐: Group Theoretical Analysis of Five-fold Symmetric Tilings

The projections of the lattices may be used as models of quasicrystals, and the particular affine extension of the H2 symmetry as a subgroup of A4, discussed in this work, presents a different perspective to 5-fold symmetric quasicrystallography. Affine H2 is obtained as the subgroup of affine A4. The infinite discrete group with local dihedral symmetry of order 10 operates on the Coxeter plane of the root and weight lattices of A4 whose Voronoi cells tessellate the 4D Euclidean space possessing the affine A4 symmetry. Facets of the Voronoi cells of the root and weight lattices are identified. Four adjacent rhombohedral facets of the Voronoi cell V(0) of A4 project into the decagonal orbit of H2 as thick and thin rhombuses, where long diagonals of the rhombohedra serve as reflection line segments of the reflection operators of H2. It is shown that the thick and thin rhombuses constitute the finite fragments of the tiles of the Coxeter plane with the action of the affine H2 symmetry. Projection of the Voronoi cell of the weight lattice onto the Coxeter plane tessellates the plane with four different tiles: thick and thin rhombuses with different edge lengths obtained from the projection of the square faces and two types of hexagons obtained from the projection of the hexagonal faces of the Voronoi cell.