[Paper Review] Dynamics and topology of the Hat family of tilings

Michael Baake, Franz Gähler+1 more|arXiv (Cornell University)|May 9, 2023

Quasicrystal Structures and Properties7 citations

TL;DR

This paper establishes that the Hat tiling and its shape-deformed variants form a 4-dimensional family of topologically conjugate dynamical systems, with a self-similar tiling (CAP tiling) arising from a 2D Euclidean cut-and-project scheme. The CAP tiling has a pure-point dynamical spectrum, and all members of the Hat family share the same topology and dynamics despite geometric deformations, revealing a deep structural invariance in aperiodic monotiling systems.

ABSTRACT

The recently discovered Hat tiling 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.

Motivation & Objective

• To investigate the dynamical and topological invariance of the Hat tiling family under continuous geometric deformations.
• To identify a self-similar tiling within the Hat family that admits a cut-and-project representation with a 2D Euclidean internal space.
• To analyze how shape deformations preserve the pure-point dynamical spectrum and topological conjugacy across the family.
• To clarify the role of reflection symmetry and the distinction between enantiomorphic LI classes in the context of monotiling.
• To explore the implications of Euclidean internal space for reprojection and the existence of infinite monotile families.

Proposed method

• Construct the CAP tiling as a self-similar element of the Hat family via a cut-and-project scheme with 2D Euclidean internal space.
• Use the theory of model sets and cut-and-project schemes to derive the pure-point dynamical spectrum of the CAP tiling.
• Demonstrate that all shape-deformed versions of the Hat tiling are topologically conjugate to the CAP tiling under the action of GL₂(ℝ), preserving dynamics.
• Analyze the deformation space using complex parameters to encode shape changes, with real parameters preserving reflection symmetry.
• Apply the concept of mutual local derivability (MLD) to classify the moduli space of shape deformations, showing it is 8-dimensional.
• Use spectral analysis to show that the spectrum of deformed tilings is rotated relative to the original, with specific angular conditions for alignment to 30° multiples.

Experimental results

Research questions

• RQ1 Does the entire Hat tiling family, under continuous shape deformation, preserve topological conjugacy and pure-point spectrum?
• RQ2 Can a self-similar tiling within the Hat family be constructed via a 2D Euclidean cut-and-project scheme?
• RQ3 Are there infinite families of monotiles arising from reprojection of the same internal space structure?
• RQ4 What is the role of reflection symmetry in distinguishing LI classes within the Hat family?
• RQ5 Can a disk-like chiral monotile enforce a unique, aperiodic LI class using only rotations and translations?

Key findings

• The moduli space of shape deformations of the Hat tiling, modulo MLD equivalence, is 8-dimensional, with 4 dimensions from rigid linear transformations and 4 from topological conjugacies.
• All tilings in the Hat family are topologically conjugate under the action of GL₂(ℝ), meaning their translation dynamics are equivalent up to linear transformation.
• The CAP tiling, a self-similar member of the family, arises from a 2D Euclidean cut-and-project scheme and has a pure-point dynamical spectrum.
• The spectrum of the CAP tiling is explicitly computed and found to be rotated by the argument of (α + iβ(φ − ξ)) relative to the original tiling axes.
• For the spectrum to align with 30°-multiples, two specific ratios of α and β are identified: β = √3 α (Turtle shape) and β ≈ 0.136α (Chevron-like shape), both yielding pure-point spectra.
• The two LI classes of the Turtle and Chevron shapes are not topologically conjugate, despite having the same spectrum, due to chirality and reflection symmetry breaking.