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arXiv · 2606.24693

An aperiodic set of Wang tiles for every quadratic irrational

Jarkko Kari·Sébastien Labbé·Pieter Mostert
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Abstract

We propose a sufficient condition for the non-periodicity of a set of Wang tiles. It applies to sets of Wang tiles whose tiles have vertical or horizontal stripes. The proof is based on a geometric argument involving a quadrilateral circumscribed to a parabola from which we conclude the irrationality of the densities of the vertical and horizontal stripes. We apply the sufficient condition to propose new proofs of non-periodicity of known sets of Wang tiles, including an encoding of Penrose tilings into 24 Wang tiles and the family of metallic mean Wang tiles. Conversely, for every pair $(\alpha,\beta)\in[0,1]^2$ of irrational numbers in the same quadratic number field, we construct a finite aperiodic set of Wang tiles with stripes that admits a valid tiling whose density of vertical stripes is $\alpha$ and density of horizontal stripes is $\beta$.

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Jarkko Kari, Sébastien Labbé, Pieter Mostert. 2026-06-23. An aperiodic set of Wang tiles for every quadratic irrational. https://arxiv.org/abs/2606.24693

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