The orthogonal projections of the Voronoi and Delone cells of root lattice An 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 an 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.
- Coxeter-Weyl groups
- Tilings by rhombi and triangles
- Voronoi and Delone cells
ASJC Scopus subject areas
- Computer Science (miscellaneous)
- Chemistry (miscellaneous)
- Physics and Astronomy (miscellaneous)