Geometrical Penrose Tilings are characterized by their 1-atlas

Abstract

Rhombus Penrose tilings are tilings of the plane by two decorated rhombi such that the decoration match at the junction between two tiles (like in a jigsaw puzzle). In dynamical terms, they form a tiling space of finite type. If we remove the decorations, we get, by definition, a sofic tiling space that we here call geometrical Penrose tilings. Here, we show how to compute the patterns of a given size which appear in these tilings by two different method: one based on the substitutive structure of the Penrose tilings and the other on their definition by the cut and projection method. We use this to prove that the geometrical Penrose tilings are characterized by a small set of patterns called vertex-atlas, i.e., they form a tiling space of finite type. Though considered as folk, no complete proof of this result has been published, to our knowledge.

Figures (28)

• Figure 1: The Penrose rhombus tiles (up to isometry): all edges are of length $1$, the thin rhombus has angles $\tfrac{\pi}{5}$ and $\tfrac{4\pi}{5}$, the fat rhombus has angles $\tfrac{2\pi}{5}$ and $\tfrac{3\pi}{5}$. In total there are 20 decorated tiles up to translation, and 10 undecorated tiles up to translation.
• Figure 2: A random tiling with the thin and fat rhombus.
• Figure 3: A geometrical Penrose tiling, i.e., a Penrose tiling where arrows or coloured-arcs have been removed. Thin rhombi are coloured in blue to improve the readability.
• Figure 4: A $1$-map centered on the red point, with its boundary emphasized. The numbers give the distance to the red point.
• Figure 5: The $0$-atlas of Penrose tilings with coloured arcs (up to isometry), the names of the patterns come from grunbaum1987.

Theorems & Definitions (16)

• Definition 1: $k$-map and $k$-atlas
• Theorem 1
• Proposition 1
• Lemma 1: solomyak1998, Linear Recurrence
• Remark 1
• Lemma 2: Linear recurrence factor for Penrose tilings
• Remark 2
• proof
• Definition 2
• Definition 3