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

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 $n\ge 1$, we show that the Wang shift ${\mathrm{\Omega }}_n$, 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 $\{{\mathrm{\Omega }}_n{\}}_{n\ge 1}$ is a family of metallic mean Wang shifts, since the inflation factor of the self-similarity of ${\mathrm{\Omega }}_n$ 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.