The Construction of Self-Similar Tilings

Abstract

We give a construction of a self-similar tiling of the plane with any prescribed expansion coefficient $λ\in\C$ (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 $2π/m$-rotational symmetry group and expansion $λ$ if and only if either $λ$ or $λe^{2πi/m}$ is a complex Perron number for which $e^{2πi/m}$ is in $\Q[λ]$, respectively $Q[λe^{2πi/m}]$.