From affine A₄ to affine H₂: group-theoretical analysis of fivefold symmetric tilings

The projections of 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 on fivefold 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. The structure of the local dihedral symmetry H₂ fixing a particular point on the Coxeter plane is determined.