Abstract
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.
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Cannon, J., Floyd, W. & Parry, W. Combinatorially Regular Polyomino Tilings. Discrete Comput Geom 35, 269–285 (2006). https://doi.org/10.1007/s00454-005-1205-2
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DOI: https://doi.org/10.1007/s00454-005-1205-2