Aperiodic Order: The Mathematics of Systems of Approximate Symmetry

Abstract

Symmetry is frequently exploited in Mathematics, but there are many situations in which systems exhibit long-range recurrence without precise periodic repetition. A simple example is given by a coding of an irrational circle rotation. With Shechtman’s discovery of quasicrystals – physical materials with long-range order but also rotational symmetry precluding the standard periodicity of usual crystals – it seems that ‘‘aperiodically ordered’’ patterns can appear in nature too. In this talk I will introduce the field of Aperiodic Order, which investigates intriguing infinite idealisations of such patterns. A prototypical family of examples is given by Penrose’s famous rhomb, or kite and dart tilings. I will then explain what sorts of mathematical structures can be introduced to systemise their study. I will focus on the construction of the tiling space of an aperiodic pattern, through which one may construct fundamental invariants using standard tools from Algebraic Topology.