Geometrical Penrose Tilings are characterized by their 1-atlas

TL;DR

This paper demonstrates that geometrical Penrose tilings can be characterized by a finite set of local patterns called the vertex-atlas, using methods based on substitution and cut-and-projection techniques.

Contribution

It provides a complete proof that geometrical Penrose tilings are characterized by a finite set of patterns, filling a gap in the existing literature.

Findings

Computed pattern sets using substitution and cut-and-projection methods

Proved geometrical Penrose tilings are of finite type characterized by a vertex-atlas

Established a new approach to analyze tilings without decorations

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.