When is a cut and project set substitutional?

TL;DR

This paper investigates the conditions under which cut and project sets, known for their aperiodic order, can also be described by substitution rules, linking geometric construction with combinatorial substitution methods.

Contribution

It characterizes the property of cut and project data that determines when these sets can be equivalently described by substitution rules in Euclidean spaces.

Findings

Identifies a specific property of the cut and project data that implies substitutional structure.

Provides criteria to distinguish substitutional cut and project sets from purely aperiodic ones.

Bridges the gap between geometric and combinatorial descriptions of aperiodic patterns.

Abstract

Cut and project sets are obtained by projecting an irrational slice through a lattice to a lower dimensional subspace. Under standard conditions, the resulting pattern has no translational periods even though it retains some regularity of the lattice. Cut and project sets are one of the archetypical examples of patterns featuring aperiodic order, the other construction methods being by substitution and matching rules. Many early examples of aperiodic tilings, including the famous Penrose and Ammann--Beenker tilings, have a description from all of these methods. In this article we answer the following question, in the case of a Euclidean total space: what property of the cut and project data characterises when the resulting cut and project sets may also be defined by a substitution rule?