Sturmian lattices and Aperiodic tile sets
Fuente:
arXiv
Salvato in:
| Autori principali: | , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2025
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866915728872964096 |
|---|---|
| author | Akiyama, Shigeki Hamada, Tadahisa Ito, Katsuki |
| author_facet | Akiyama, Shigeki Hamada, Tadahisa Ito, Katsuki |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_19362 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Sturmian lattices and Aperiodic tile sets Akiyama, Shigeki Hamada, Tadahisa Ito, Katsuki Combinatorics Dynamical Systems Metric Geometry Number Theory 05B45, 68R15, 52C23, 37B52 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. |
| title | Sturmian lattices and Aperiodic tile sets |
| topic | Combinatorics Dynamical Systems Metric Geometry Number Theory 05B45, 68R15, 52C23, 37B52 |
| url | https://arxiv.org/abs/2506.19362 |