On the Structure of Multiple Translational Tilings by Polygonal Regions

Abstract

We consider polygons with the following ``pairing property'': for each edge of the polygon there is precisely one other edge parallel to it. We study the problem of when such a polygon K tiles multiply the plane when translated at the locations Λ , where Λ is a multiset in the plane. The pairing property of K makes this question particularly amenable to Fourier analysis. As a first application of our approach we establish a necessary and sufficient condition for K to tile with a given lattice Λ . (This was first found by Bolle for the case of convex polygons—notice that all convex polygons that tile, necessarily have the pairing property and, therefore, our theorems apply to them.) Our main result is a proof that a large class of such polygons tile multiply only quasi-periodically, which for us means that Λ must be a finite union of translated two-dimensional lattices in the plane. For the particular case of convex polygons we show that all convex polygons which are not parallelograms tile multiply only quasi-periodically, if at all.