Combinatorially Regular Polyomino Tilings

Let T be a regular tiling of R which has the origin 0 as a vertex, and suppose that φ : R → R 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.