Classification of disordered tilings

Abstract

A tilingT is a disordered realization of a periodic tilingP with symmetry group Γ if we can map the complement of a compact set ofT onto the quotientP/Γ in such a way that this map respects the features of the tilingT andP. We show that the global type of a 2-dimensional tilingT is determined by the periodic tilingP it is a disordered realization of, a conjugacy class of Γ which can be associated toT and a winding number. In some cases, we need in addition some kind of orientation. For higher-dimensional tilings of spaces which are simply connected at infinity, e.g. ℝⁿ withn≥3, the associated periodic tiling alone is sufficient.