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

For every positive integer n , we introduce a set Tₙ made of (n+3)² Wang tiles (unit squares with labeled edges). We represent a tiling by translates of these tiles as a configuration Z²→ Tₙ . A configuration is valid if the common edge of adjacent tiles has the same label. For every n≥ 1 , we show that the Wang shift Ω ₙ , defined as the set of valid configurations over the tiles Tₙ , is self-similar, aperiodic and minimal for the shift action. We say that \Ω ₙ≥ 1 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 x²-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 T₁ is equivalent to the Ammann aperiodic set of 16 Wang tiles.