A note on tilings and translation surfaces

Abstract

Consider a tiling 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 T, there exists a tiling of a (compact) translation surface made with copies up to translation of some of the polygons used to construct T. Furthermore, when 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 T; each of these patches contain copies of all the polygons used to construct T.