Combinatorial problems of (quasi-)crystallography

Abstract

Several combinatorial problems of (quasi-)crystallography are reviewed with special emphasis on a unified approach, valid for both crystals and quasicrystals. In particular, we consider planar sublattices, similarity sublattices, coincidence sublattices, their module counterparts, and central and averaged shelling. The corresponding counting functions are encapsulated in Dirichlet series generating functions, with explicit results for the triangular lattice and the twelvefold symmetric shield tiling. Other combinatorial properties are briefly summarised.

12 pages, 2 PostScript figures, LaTeX using vch-book.cls

References in corpus (4)

• Planar coincidences for N-fold symmetry
• Solution of the coincidence problem in dimensions $d\le 4$
• Colourings of planar quasicrystals
• Coordination Sequences and Critical Points

Cited by in corpus (7)

• Homometric model sets and window covariograms
• Bravais colourings of planar modules with N-fold symmetry
• Similarity and Coincidence Isometries for Modules
• Q-Dependent Susceptibility in Z-Invariant Pentagrid Ising Model
• Realizability of integer sequences as differences of fixed point count sequences
• Ramanujan and Eckford Cohen totients from Visible Point Identities
• Dirichlet convolution and enumeration of pyramid polycubes