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 $n\geq 1$ , 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 $\{{\Omega }_n\}_{n\geq 1}$ 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.