Prototiles and Tilings from Voronoi and Delone Cells of the Root Lattice An

Abstract

The orthogonal projections of the Voronoi and Delone cells of root lattice A n onto the Coxeter plane display various rhombic and triangular prototiles including thick and thin rhombi of Penrose, Amman–Beenker tiles, Robinson triangles, and Danzer triangles to name a few. We point out that the symmetries representing the dihedral subgroup of order 2 h involving the Coxeter element of order h = n + 1 of the Coxeter–Weyl group a n play a crucial role for h -fold symmetric tilings of the Coxeter plane. After setting the general scheme we give samples of patches with 4-, 5-, 6-, 7-, 8-, and 12-fold symmetries. The face centered cubic (f.c.c.) lattice described by the root lattice A 3 , whose Wigner–Seitz cell is the rhombic dodecahedron projects, as expected, onto a square lattice with an h = 4 -fold symmetry.