The Construction of Self-Similar Tilings

Details The Construction of Self-Similar Tilings The Construction of Self-Similar Tilings

1995-05-30

arxiv.org/abs/math/9505210v1

Mathematics

Metric Geometry

Scientific paper

We give a construction of a self-similar tiling of the plane with any prescribed expansion coefficient $\lambda\in\C$ (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 $2\pi/m$-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 $\Q[\lambda]$, respectively $Q[\lambda e^{2\pi i/m}]$.

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