When is a polytopal cut and project set a substitution pattern?

Abstract

The two main methods of constructing aperiodically ordered patterns are by substitution and the cut and project method. Many interesting examples in Aperiodic Order, such as the Penrose tilings and Ammann–Beenker tilings, are both substitution patterns and cut and project sets with polytopal windows. In this talk I will discuss recent work with Edmund Harriss and Henna Koivusalo that gives a simple set of necessary and sufficient conditions for a cut and project scheme with polytopal window to define patterns that are also generated by substitution. This builds off previous results of Harriss and Lamb in this area. A key new feature is a simplification of some core definitions, in particular what is meant by a ‘substitution pattern’. We use a definition similar to that of a LIDS (local inflation deflation symmetry), although one that applies to all elements of the hull. This definition covers all examples of interest (whether they are stone or non-stone tile substitutions, or Delone set substitutions), still implies the main useful properties of being generated by substitution (such as having linear repetitivity in the self-similar case) and allows for simpler proofs of our main results.