## Combinatorially regular polyomino tilings

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

### Abstract

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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