Combinatorially regular polyomino tilings

J. W. Cannon, W. J. Floyd, and W. R. Parry


Let $\cT$ be a regular tiling of $\bR^2$ which has the origin $0$ as a vertex, and suppose that $\varphi\co \bR^2 \to \bR^2$ is a homeomorphism such that i) $\varphi(0)=0$, ii) the image under $\varphi$ of each tile of $\cT$ is a union of tiles of $\cT$, and iii) the images under $\varphi$ of any two tiles of $\cT$ are equivalent by an orientation-preserving isometry which takes vertices to vertices. It is proved here that there is a subset $\Lambda$ of the vertices of $\cT$ such that $\Lambda$ is a lattice and $\varphi|_{\Lambda}$ is a group homomorphism. The tiling $\varphi(\cT)$ is a tiling of $\bR^2$ 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 $\varphi(\cT)$ is conjugate to a self-similar tiling.

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