Aperiodic Wang tiles associated with metallic means.

RESUMEN: A Penrose tiling consists of two polygonal tiles whose frequency ratio is equal to the golden ratio. Similarly, tilings by the aperiodic monotile discovered in 2023 by David Smith are such that the ratio of the frequencies of the two orientations of the monotile is equal to the fourth power of the golden ratio. The structure of Jeandel-Rao tilings discovered in 2015 is also explained using the golden ratio. We know of aperiodic tilings that are not related to the golden ratio. However, the characterization of possible numbers for such ratios is a question, posed as early as 1992 by Ammann, Grünbaum and Shephard, which is still open today.

We introduce a new family of sets of aperiodic Wang tiles (unit squares with labeled edges). The family extends the relationship between quadratic integers and aperiodic tilings beyond the ubiquitous golden ratio, as its dynamics involves the positive root of the polynomial x² − nx − 1. This root is sometimes called the n-th metallic mean, and in particular, the golden ratio when n = 1 and the silver ratio when n = 2. Details can be found in the prepublications available at https://arxiv.org/abs/2312.03652 (Part I) and https://arxiv.org/abs/2403.03197 (Part II).

Posted on Nov 11, 2024 in Dynamical Systems, Seminars