From Affine $A_4$ to Affine $H_2$: Group Theoretical Analysis of Five-fold Tilings

The projections of the lattices, may be used as models of quasicrystals, and the particular affine extension of the $H_2$ symmetry as a subgroup of $A_4$, discussed in the work, presents a different perspective to 5-fold symmetric quasicrystallography. Affine $H_2$ is obtained as the subgroup of the affine $A_4$. The infinite group with local dihedral symmetry of order 10 operates on the Coxeter plane of the root and weight lattices of $A_4$ whose Voronoi cells tessellate the 4D Euclidean space possessing the affine $A_4$ symmetry. It is shown that the projection of the Voronoi cell of the root lattice tiles the Coxeter plane with thick and thin rhombuses with the action of the affine $H_2$ 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. Structure of the local dihedral symmetry $H_2$ fixing a particular point on the Coxeter plane is determined.