On the combinatorial characterization of quasicrystals

In various fields of materials science, many interesting two-dimensional (2D) and three-dimensional (3D) structures (fullerenes, nanotubes, frothes, metal foams, polycrystals and, notably, various quasicrystals) can be considered as finite or infinite cellular systems. For a cell TT of a given 2D cellular system, we write n(T)n(T) to denote the number of sides of TT, and m(T)m(T) to denote the average number of sides of the neighbours of TT. In the 3D case, we replace sides by faces in the definition. D. Weaire first observed, for trivalent random tilings of the plane, that 〈n⋅m〉=〈n2〉〈n⋅m〉=〈n2〉, where 〈⋅〉〈⋅〉 stands for the expected value. Following his discovery, the Weaire sum rule has been proved for various tilings of a sphere or a torus, and for periodic tilings of the plane or space. In this paper we extend the Weaire sum rule to quasiperiodic tilings of the plane or space. Actually, the method of this paper yields the Weaire sum rule for tilings of any compact surface or three-manifolds as well.