Group-theoretical analysis of aperiodic tilings from projections of higher-dimensional lattices Bn

Mehmet Koca, Nazife Ozdes Koca*, Ramazan Koc

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

6 Citations (Scopus)


A group-theoretical discussion on the hypercubic lattice described by the affine Coxeter-Weyl group Wa (Bn) is presented. When the lattice is projected onto the Coxeter plane it is noted that the maximal dihedral subgroup Dh of W(Bn) with h = 2n representing the Coxeter number describes the h-fold symmetric aperiodic tilings. Higher-dimensional cubic lattices are explicitly constructed for n = 4, 5, 6. Their rank-3 Coxeter subgroups and maximal dihedral subgroups are identified. It is explicitly shown that when their Voronoi cells are decomposed under the respective rank-3 subgroups W(A 3), W(H 2) × W(A 1) and W(H 3) one obtains the rhombic dodecahedron, rhombic icosahedron and rhombic triacontahedron, respectively. Projection of the lattice B 4 onto the Coxeter plane represents a model for quasicrystal structure with eightfold symmetry. The B 5 lattice is used to describe both fivefold and tenfold symmetries. The lattice B 6 can describe aperiodic tilings with 12-fold symmetry as well as a three-dimensional icosahedral symmetry depending on the choice of subspace of projections. The novel structures from the projected sets of lattice points are compatible with the available experimental data.

Original languageEnglish
Pages (from-to)175-185
Number of pages11
JournalActa Crystallographica Section A: Foundations and Advances
Publication statusPublished - Mar 1 2015
Externally publishedYes


  • Coxeter-Weyl groups
  • Lattices
  • aperiodic tilings
  • cut-and-project technique
  • quasicrystallography
  • strip projection

ASJC Scopus subject areas

  • Structural Biology
  • Biochemistry
  • Materials Science(all)
  • Condensed Matter Physics
  • Physical and Theoretical Chemistry
  • Inorganic Chemistry


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