Prototiles and Tilings from Voronoi and Delone Cells of the Root Lattice A(n)


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Ozdes Koca N., Al-Siyabi A., Koca M., KOÇ R.

SYMMETRY-BASEL, cilt.11, 2019 (SCI İndekslerine Giren Dergi) identifier identifier

  • Cilt numarası: 11 Konu: 9
  • Basım Tarihi: 2019
  • Doi Numarası: 10.3390/sym11091082
  • Dergi Adı: SYMMETRY-BASEL

Özet

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 A(3), whose Wigner-Seitz cell is the rhombic dodecahedron projects, as expected, onto a square lattice with an h = 4-fold symmetry.