Metallic mean Wang tiles I: self-similarity, aperiodicity and minimality
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| Format: | Preprint |
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2023
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| _version_ | 1866908556894142464 |
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| author | Labbé, Sébastien |
| author_facet | Labbé, Sébastien |
| contents | For every positive integer $n$, we introduce a set $\mathcal{T}_n$ made of $(n+3)^2$ Wang tiles (unit squares with labeled edges). We represent a tiling by translates of these tiles as a configuration $\mathbb{Z}^2\to\mathcal{T}_n$. A configuration is valid if the common edge of adjacent tiles has the same label. For every $n\geq1$, we show that the Wang shift $Ω_n$, defined as the set of valid configurations over the tiles $\mathcal{T}_n$, is self-similar, aperiodic and minimal for the shift action. We say that $\{Ω_n\}_{n\geq1}$ is a family of metallic mean Wang shifts, since the inflation factor of the self-similarity of $Ω_n$ is the positive root of the polynomial $x^2-nx-1$. This root is sometimes called the $n$-th metallic mean, and in particular, the golden mean when $n=1$, and the silver mean when $n=2$. When $n=1$, the set of Wang tiles $\mathcal{T}_1$ is equivalent to the Ammann aperiodic set of 16 Wang tiles. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2312_03652 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Metallic mean Wang tiles I: self-similarity, aperiodicity and minimality Labbé, Sébastien Dynamical Systems Combinatorics Metric Geometry Number Theory 52C23 (Primary) 37B51, 37B05, 11B39 (Secondary) For every positive integer $n$, we introduce a set $\mathcal{T}_n$ made of $(n+3)^2$ Wang tiles (unit squares with labeled edges). We represent a tiling by translates of these tiles as a configuration $\mathbb{Z}^2\to\mathcal{T}_n$. A configuration is valid if the common edge of adjacent tiles has the same label. For every $n\geq1$, we show that the Wang shift $Ω_n$, defined as the set of valid configurations over the tiles $\mathcal{T}_n$, is self-similar, aperiodic and minimal for the shift action. We say that $\{Ω_n\}_{n\geq1}$ is a family of metallic mean Wang shifts, since the inflation factor of the self-similarity of $Ω_n$ is the positive root of the polynomial $x^2-nx-1$. This root is sometimes called the $n$-th metallic mean, and in particular, the golden mean when $n=1$, and the silver mean when $n=2$. When $n=1$, the set of Wang tiles $\mathcal{T}_1$ is equivalent to the Ammann aperiodic set of 16 Wang tiles. |
| title | Metallic mean Wang tiles I: self-similarity, aperiodicity and minimality |
| topic | Dynamical Systems Combinatorics Metric Geometry Number Theory 52C23 (Primary) 37B51, 37B05, 11B39 (Secondary) |
| url | https://arxiv.org/abs/2312.03652 |