Double Tiles

Which polygons admit two (or more) distinct lattice tilings of the plane? We call such polygons double tiles. It is well-known that a lattice tiling is always combinatorially isomorphic either to a grid of squares or to a grid of regular hexagons. We focus on the special case of the double tile problem where both tilings are in the square class. For this special case, we give an explicit description of all double tiles. We establish the result for polyominoes first; then, with little additional effort, we extend the proof to general polygons. Central to the description is a certain finite set of transformations which we apply iteratively to a base shape in order to obtain one family of "fractal-like" polyominoes. The double tiles are then given by these polyominoes together with particular "deformations" of them.