Prototiles and tilings from voronoi and delone cells of the root lattice A_n

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 2h 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 A₃, whoseWigner-Seitz cell is the rhombic dodecahedron projects, as expected, onto a square lattice with an h = 4-fold symmetry.