Affine Dihedral Subgroups of Higher Dimensional Cubic Lattices $\mathbb{Z}^n$ and Quasicrystallography

TL;DR

This paper explores the affine dihedral subgroups of higher dimensional cubic lattices and their projections onto the Coxeter plane, revealing connections to quasicrystal structures with various rotational symmetries.

Contribution

It introduces a new perspective on quasicrystallography by analyzing affine dihedral subgroups within higher dimensional cubic lattices and their projections onto the Coxeter plane.

Findings

Projection of B_3 yields hexagonal lattice

Projection of B_4 describes 8-fold Ammann-Beenker quasicrystal

Projection of B_5 describes 10-fold quasicrystal with rhombi

Abstract

Quasicrystals described as the projections of higher dimensional cubic lattices, and the particular affine extensions of the dihedral group $I_2(h)$ of order $2h$, $h=2n$ being the Coxeter number, as a subgroup of affine $B_n$ offers a different perspective to $h$-fold symmetric quasicrystallography. Affine $I_2(h)$ is constructed as the subgroup of the affine $B_n$, the symmetry of the cubic lattice $\mathbb{Z}^n$. The infinite discrete group with local dihedral symmetry of order $2h$ operates on the concentric h-gons obtained by projecting the Voronoi cell of the cubic lattice with $2^n$ vertices onto the Coxeter plane. Voronoi cells tile the space facet to facet, consequently, leading to the tilings of the Coxeter plane with some overlaps of the rhombic tiles. It is noted that the projected Voronoi cell is the overlap of $h$ copy of the $h$-gons tiled with some rhombi and rotated by the angle $2\pi/h$. After a general discussion on the lattice $\mathbb{Z}^n$ with the affine symmetry $\tilde B_n$ and its affine dihedral subgroup $\tilde I_2(h)$ its projection onto the Coxeter plane has been worked out with some examples. The cubic lattices with affine symmetry $\tilde B_n$ $(n=1,2,3,4,5)$ have been presented and shown that the projection of the lattice $B_3$ leads to the hexagonal lattice, the projection of the lattice $B_4$ describes the Ammann-Beenker quasicrystal lattice with 8-fold local symmetry and the projection of the lattice $B_5$ describes a quasicrystal structure with local 10-fold symmetry with thick and thin rhombi. It is then straight forward to show that the projections of the cubic lattices with even higher dimensions onto the Coxeter plane may lead to the quasicrystal structures with 12-fold, 18-fold symmetries and so on.