A note on tilings and translation surfaces

Abstract

Consider a tiling $\mathcal T$ of the two-dimensional Euclidean space made with copies up to translation of a finite number of polygons meeting each other full edge to full edge. In this paper, we prove that, associated with $\mathcal T$, there exists a tiling of a (compact) translation surface made with copies up to translation of some of the polygons used to construct $\mathcal T$. Furthermore, when $\mathcal T$ is repetitive, there exists a tiling of a translation surface, made with copies up to translation of arbitrarily large polygons chosen in a finite collection of patches of $\mathcal T$; each of these patches contain copies of all the polygons used to construct $\mathcal T$.