Resurrecting the space group of an aperiodic pattern

Abstract

A periodic pattern of Euclidean space is completely determined, up to “locally reversible redecorations”, by its space group of global symmetries. For an aperiodically ordered pattern the group of global symmetries is not a particularly enlightening object: the interesting thing about aperiodically ordered patterns is that they can have a rich structure of internal symmetries whilst having very few (if any) global symmetries. In this talk I shall discuss recent work with John Hunton which expresses the space group of a periodic pattern via a topological route. These rotational topological invariants can be applied to periodic as well as non-periodic patterns, which suggests that they are one approach to defining the analogue of the space group, or of its invariants, for an aperiodic pattern.