Metallic mean Wang tiles I: self-similarity, aperiodicity and minimality

Abstract

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 $\Omega_n$, 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 $\{\Omega_n\}_{n\geq1}$ is a family of metallic mean Wang shifts, since the inflation factor of the self-similarity of $\Omega_n$ 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.