Dynamics and topology of the Hat family of tilings

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.

• Publication year

  2023

• Venue

  Israel Journal of Mathematics

• Publication date

  2023-05-09

• Fields of study

  Mathematics

• Identifiers
  DOI 10.1007/s11856-025-2780-8arXiv 2305.05639
• External record

  Open on Semantic Scholar

• Source metadata

  Semantic Scholar

• No claims are published for this paper.

• No concepts are published for this paper.

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