Combinatorially Regular Polyomino Tilings

Let T be a regular tiling of ℝ² which has the origin 0 as a vertex, and suppose that φ: ℝ² → ℝ² is a homeomorphism such that (i) φ(0)=0, (ii) the image under φ of each tile of T is a union of tiles of T, and (iii) the images under φ of any two tiles of T are equivalent by an orientation-preserving isometry which takes vertices to vertices. It is proved here that there is a subset Λ of the vertices of T such that Λ is a lattice and φ|_Λ is a group homomorphism. The tiling φ(T) is a tiling of ℝ by polyiamonds, polyominos, or polyhexes. These tilings occur often as expansion complexes of finite subdivision rules. The above theorem is instrumental in determining when the tiling φ(T) is conjugate to a self-similar tiling.