Cut and project sets: complexity, repetitivity and self-similarity

Abstract

Many famous examples from Aperiodic Order, such as the Penrose tilings, Ammann-Beenker tilings or tilings of the recently discovered hat monotile, turn out to be constructable from the cut and project method. Roughly speaking, a cut and project scheme takes an ‘irrational slice’ of a periodic pattern (a lattice) in a higher dimensional space, producing a structure which is no longer periodic but is still ‘ordered’. In this talk I will introduce central concepts, such as this, from the field of Aperiodic Order, including how these patterns can be studied from the perspective of Dynamical Systems. I will then explain how one may determine properties of cut and project sets which have polytopal acceptance windows: the growth rate of their patch counting functions (or ‘complexity’), whether or not they have linear repetitivity and whether or not they are ‘self-similar’, that is, generated from a substitution rule.