Sturmian lattices and Aperiodic tile sets
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.
Keywords
Cite
@article{arxiv.2506.19362,
title = {Sturmian lattices and Aperiodic tile sets},
author = {Shigeki Akiyama and Tadahisa Hamada and Katsuki Ito},
journal= {arXiv preprint arXiv:2506.19362},
year = {2026}
}
Comments
Major update. We give a classification of Sturmian lattices including their hierarchical structure. The construction of aperiodic tile sets is made explicit for any quadratic expansion constants