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 $\beta$.
From: Sébastien Labbé [view email]
[v1]
Tue, 23 Jun 2026 15:22:15 UTC (119 KB)
此内容由惯性聚合(RSS阅读器)自动聚合整理，仅供阅读参考。 原文来自 — 版权归原作者所有。