Sturmian lattices and Aperiodic tile sets

TL;DR

This paper presents an explicit algorithm for constructing aperiodic tile sets using Sturmian words with quadratic slopes, enabling the creation of infinitely many such sets linked to units in real quadratic fields.

Contribution

It introduces Sturmian lattices and their classification, and demonstrates how to generate aperiodic tile sets based on these structures for quadratic irrational slopes.

Findings

Constructed infinitely many aperiodic tile sets for quadratic slopes

Classified Sturmian lattices and their properties

Linked tile set construction to units in real quadratic fields

Abstract

We give an explicit algorithm to construct aperiodic tile sets based on Sturmian words of quadratic slopes. The method works for any quadratic irrational slope, and we can produce infinitely many aperiodic tile sets whose underlying scaling constant is a unit of any real quadratic field. There are two key ingredients in our construction. The first one is ``Sturmian lattices''; an interesting grid structure generated by Sturmian words that emerged in an aperiodic monotile called Smith Turtle. We shall give a classification of Sturmian lattices. The second is the bounded displacement equivalence of Delone sets, which plays a central role in this construction.