The Construction of Self-Similar Tilings
Richard Kenyon
We give a construction of a self-similar tiling of the plane with any prescribed expansion coefficient $\lambda\in{\Bbb C}$ (satisfying the necessary algebraic condition of being a complex Perron number).
For any integer $m>1$ we show that there exists a self-similar tiling with $2\pi/m$-rotational symmetry group and expansion $\lambda$ if and only if either $\lambda$ or $\lambda e^{2\pi i/m}$ is a complex Perron number for which $e^{2\pi i/m}$ is in ${\Bbb Q}[\lambda]$, respectively ${\Bbb Q}[\lambda e^{2\pi i/m}]$.