Coordination sequences of crystals are of quasi-polynomial type

The coordination sequence of a graph measures how many vertices the graph has at each distance from a fixed vertex and is a generalization of the coordination number. Here it is proved that the coordination sequence of the graph obtained from a crystal is of quasi-polynomial type, as had been postulated by Grosse-Kunstleve et al. [Acta Cryst. (1996), A52, 879-889].

Coordination shells and coordination numbers of the vertex graph of the Ammann-Beenker tiling

The vertex graph of the Ammann-Beenker tiling is a well-known quasiperiodic graph with an eightfold rotational symmetry. The coordination sequence and coordination shells of this graph are studied. It is proved that there exists a limit growth form for the vertex graph of the Ammann-Beenker tiling. This growth form is an explicitly calculated regular octagon. Moreover, an asymptotic formula for the coordination numbers of the vertex graph of the Ammann-Beenker tiling is also proved.

A coloring-book approach to finding coordination sequences

An elementary method is described for finding the coordination sequences for a tiling, based on coloring the underlying graph. The first application is to the two kinds of vertices (tetravalent and trivalent) in the Cairo (or dual-3².4.3.4) tiling. The coordination sequence for a tetravalent vertex turns out, surprisingly, to be 1, 4, 8, 12, 16, …, the same as for a vertex in the familiar square (or 4⁴) tiling. The authors thought that such a simple fact should have a simple proof, and this article is the result. The method is also used to obtain coordination sequences for the 3².4.3.4, 3.4.6.4, 4.8², 3.12² and 3⁴.6 uniform tilings, and the snub-632 tiling. In several cases the results provide proofs for previously conjectured formulas.

Coordination numbers of the vertex graph of a Penrose tiling

A new approach to study coordination shells and coordination sequences of quasiperiodic graphs is suggested. The structure of the coordination shells in the vertex graph of a Penrose tiling is described. An asymptotic formula for its coordination numbers is obtained. An essentially different behaviour of the coordination numbers for even and odd shells is revealed.