The Number of Words of a Given Length in the Planar Crystallographic Groups

Quasiperiodic plane tilings based on stepped surfaces

“…The coordination sequence is strictly connected with the layerwise growth model. It is possible to give exact formulas for coordination numbers in a periodic case (Shutov, 2005). For quasiperiodic tilings, only asymptotic formulas can be obtained (Baake & Grimm, 2003).…”

Coordination numbers of the vertex graph of a Penrose tiling

“…This conjecture completely describes the behaviour of coordination numbers in the periodic case. Various special cases of this conjecture were proved by Conway & Sloane (1997), Baake & Grimm (1997), Eon (2002) and Shutov (2005). A full proof of this conjecture is currently unknown.…”

On the growth of the wallpaper groups

“…In the oriented case, one can also define a pseudometric on the graph, and hence also a growth function. In [23], Shutov presented the growth functions of all 20 oriented graphs (some groups are presented by Coxeter and Moser with more than one system of generators and relators), using the method of the article of Zhuravlev [24]. Unfortunately, the expressions for the oriented version of γ(n) are claimed to be valid for n ≥ n 0 for some n 0 not indicated in [23].…”

“…In [23], Shutov presented the growth functions of all 20 oriented graphs (some groups are presented by Coxeter and Moser with more than one system of generators and relators), using the method of the article of Zhuravlev [24]. Unfortunately, the expressions for the oriented version of γ(n) are claimed to be valid for n ≥ n 0 for some n 0 not indicated in [23]. Also, details of the proofs are missing and the growth of Γ based on its orientation does not always coincide with the growth of Γ as an undirected graph.…”