Year: 2025
Type: article
Abstract: The recently discovered Hat tiling [18] admits a 4-dimensional family of shape deformations, including the 1-parameter family already known to yield alternate monotiles. The continuous hulls resulting from these tilings are all topologically conjugate dynamical systems, and hence have the same dynamics and topology. We construct and analyze a self-similar element of this family called the CAP tiling, and we use it to derive properties of the entire family. The CAP tiling has pure-point dynamical spectrum, which we compute explicitly, and comes from a natural cut-and-project scheme with 2-dimensional Euclidean internal space. All other members of the Hat family, in particular the original version constructed from 30-60-90 right triangles, are obtained via small modifications of the projection from this cut-and-project scheme.
Source: Israel Journal of Mathematics
Authors: Michael Baake, Franz Gähler, Lorenzo Sadun
Institutions: Bielefeld University, The University of Texas at Austin
Language: English
Subfield: Materials Chemistry
Field: Materials Science
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,290
Funder: Deutsche Forschungsgemeinschaft
OpenAlex ID: W4411964559