Pieter MostertJarkko Kari, Sébastien Labbé and I have a pre-print on the arXiv on a sufficient condition for proving non-periodicity of Wang tiles with vertical and horizontal stripes: https://arxiv.org/abs/2606.24693
There's a nice connection to quadrilaterals and the two associated families of lines that are all tangent to a fixed parabola - see https://www.geogebra.org/m/a4vmxgft
This condition applies to - among others - the set of 24 Wang tiles encoding the Penrose P3 tiles and the metallic mean family of aperiodic tiles. Here the densities of both vertical and horizontal stripes are equal to the inverses of the golden mean and the metallic means \(\frac{n+\sqrt{n^2+4}}{2}\), \(n\in \mathbb{Z}_{>0}\) , respectively.
We also apply this to a new family of aperiodic Wang tile sets, constructed using some of the ideas from Kari's set of 14 Wang tiles: https://www.sciencedirect.com/science/article/pii/0012365X9500120L. Given any irrational \(\alpha, \beta \in (0,1)\) in the same real quadratic number field, this family contains a tile set with a tiling whose vertical and horizontal stripe densities are \(\alpha\) and \(\beta\) respectively.
The construction outlined here https://mathstodon.xyz/@pieter/115532375133320451 and here https://mathstodon.xyz/@pieter/115807592751389558 also makes use of the non-periodicity argument, but is different from the one in the paper. The connection between these two constructions will be explored in a subsequent paper.
