On tilings of the plane

Abstract

The paper discusses homeomorphic types of (periodic) tilings of the plane in terms of their associated Delaney symbol. Such a symbol consists of a (finite) set D on which three involutions σ0, σ1 and σ2 act from the right such that σ0σ2=σ2σ0 and there are two maps m 01, m 12 : D satisfying certain compatibility conditions. It is shown how the barycentric subdivision of a tiling can be used to define its Delaney symbol and that the symbol characterizes the tiling up to (equivariant) homeomorphisms. Furthermore, it is shown how properties of the tiling can be recognized from corresponding properties of the symbol and how this technique can be used to enumerate various types of tilings with specific properties. If necessary, this enumeration can be done by appropriate computer programs. Among other results, we have been able to vindicate the results by Grunbaum et al., announced in [8]. Finally, some recursive enumeration formulas, based on the Delaney symbol technique, are stated.