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

Nazife Ozdes Koca*, Abeer Al-Siyabi, Mehmet Koca, Ramazan Koc

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

1 Citation (Scopus)

Abstract

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.

Original languageEnglish
Article number1082
JournalSymmetry
Volume11
Issue number9
DOIs
Publication statusPublished - 2019

Keywords

  • Coxeter-Weyl groups
  • Lattices
  • Tilings by rhombi and triangles
  • Voronoi and Delone cells

ASJC Scopus subject areas

  • Computer Science (miscellaneous)
  • Chemistry (miscellaneous)
  • Mathematics(all)
  • Physics and Astronomy (miscellaneous)

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