12-Fold Symmetric Quasicrystallography from the lattices F4, B6, and E6

Nazife O. Koca, Mehmet Koca, Ramazan Koc

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One possible way to obtain the quasicrystallographic structure is the projection of the higher dimensional lattice into 2D or 3D subspaces. In this work we introduce a general technique applicable to any higher dimensional lattice. We point out that the Coxeter number and the integers of the Coxeter exponents of a Coxeter-Weyl group play a crucial role in determining the plane onto which the lattice to be projected. The quasicrystal structures display the dihedral symmetry of order twice the Coxeter number. The eigenvectors and the corresponding eigenvalues of the Cartan matrix are used to determine the set of orthonormal vectors in nD Euclidean space which lead suitable choices for the projection subspaces. The maximal dihedral subgroup of the Coxeter-Weyl group is identified to determine the symmetry of the quasicrystal structure. We give examples for 12-fold symmetric quasicrystal structures obtained by projecting the higher dimensional lattices determined by the affine Coxeter-Weyl groups Wa(F4),Wa (B6), and Wa (E6) . These groups share the same Coxeter number h =12with different Coxeter exponents. The dihedral subgroup D12 of the Coxeter groups can be obtained by defining two generators R1 and R2 as the products of generators of the Coxeter-Weyl groups. The reflection generators R1 and R2 operate in the Coxeter planes where the Coxeter element R1 R2 of the Coxeter-Weyl group represents the rotation of order 12. The canonical projections (strip projection, equivalently, cut and project technique) of the lattices determine the nature of the quasicrystallographic structures with 12-fold symmetry as well as the crystallographic structures with 4-fold and 6-fold symmetry. We note that the quasicrystal structures obtained from the lattices Wa (F4) and Wa (B6) are compatible with some experimental results.

Original languageEnglish
Pages (from-to)1-24
Number of pages24
JournalActa Crystallographica Section A: Foundations and Advances
Issue number6
Publication statusPublished - 2014


  • Coxeter-weyl groups
  • Cut and project technique
  • Lattices
  • Quasicrystallography
  • Strip projection

ASJC Scopus subject areas

  • Structural Biology
  • Biochemistry
  • General Materials Science
  • Condensed Matter Physics
  • Physical and Theoretical Chemistry
  • Inorganic Chemistry


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