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.
From: Shigeki Akiyama [view email]
[v1]
Tue, 24 Jun 2025 06:45:23 UTC (3,544 KB)
[v2]
Tue, 8 Jul 2025 12:58:31 UTC (3,547 KB)
[v3]
Wed, 14 Jan 2026 13:15:11 UTC (9,071 KB)
[v4]
Sun, 9 Aug 2026 22:32:40 UTC (11,903 KB)
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