English

The Construction of Self-Similar Tilings

Metric Geometry 2016-09-06 v1

Abstract

We give a construction of a self-similar tiling of the plane with any prescribed expansion coefficient λ\C\lambda\in\C (satisfying the necessary algebraic condition of being a complex Perron number). For any integer m>1m>1 we show that there exists a self-similar tiling with 2π/m2\pi/m-rotational symmetry group and expansion λ\lambda if and only if either λ\lambda or λe2πi/m\lambda e^{2\pi i/m} is a complex Perron number for which e2πi/me^{2\pi i/m} is in \Q[λ]\Q[\lambda], respectively Q[λe2πi/m]Q[\lambda e^{2\pi i/m}].

Keywords

Cite

@article{arxiv.math/9505210,
  title  = {The Construction of Self-Similar Tilings},
  author = {Richard Kenyon},
  journal= {arXiv preprint arXiv:math/9505210},
  year   = {2016}
}