Aperiodic Order and Linear Repetitivity of Polytopal Cut and Project Sets

Abstract

A cut and project set is a point pattern in Euclidean space given by cutting an irrational slice of a higher dimensional lattice and then projecting it to the so-called physical space. The vertices of the Penrose Tilings are famous examples of cut and project sets. These patterns inherit some of the order of the original lattice – sometimes even rotational symmetry – but the irrationality of the slice removes any global translational symmetry, making them examples of Aperiodic Order. In this talk I will introduce the study of Aperiodic Order through some of its main examples, constructions and properties of interest. I will then present recent work, joint with Henna Koivusalo, which gives a characterisation of linear repetitivity (a signifier of high structural order) of a large class of cut and project sets in terms of Diophantine approximation properties of the original cut and project scheme.