Diffraction of the Hat and Spectre tilings and some of their relatives

TL;DR

This work derives explicit, pure-point diffraction for the Hat and Spectre monotile tilings by embedding their dynamical systems into Euclidean cut-and-project schemes and exploiting an exact Fourier cocycle for model sets with fractal windows. By using reprojections and controlled shape deformations, the authors connect the Hat to CAP tilings and the Spectre to CASPr tilings, enabling closed-form expressions for Fourier–Bohr amplitudes and diffraction intensities across multiple related tilings. The approach unifies the spectral analysis of these aperiodic monotiles, shows how topological conjugacy preserves the dynamical spectrum under reprojection, and provides practical algorithms (via cocycles) to compute diffraction with arbitrary precision. The results illuminate how window fractal boundaries influence amplitude decay and reveal the lattice-periodic structure in the deformed cases, with broad implications for understanding long-range order in aperiodic tilings and their physical diffraction signatures.

Abstract

The diffraction spectra of the Hat and Spectre monotile tilings, which are known to be pure point, are derived and computed explicitly. This is done via model set representatives of self-similar members in the topological conjugacy classes of the Hat and the Spectre tiling, which are the CAP and the CASPr tiling, respectively. This is followed by suitable reprojections of the model sets to represent the original Hat and Spectre tilings, which also allows to calculate their Fourier--Bohr coefficients explicitly. Since the windows of the underlying model sets have fractal boundaries, these coefficients need to be computed via an exact renormalisation cocycle in internal space.

Figures (20)

• Figure 1: The geometric inflation rule with prototiles of the natural lengths as induced by the substitution $\varrho$.
• Figure 2: Diffraction intensities for model sets with equal weights belonging to the substitutions $\varrho$, and $\widetilde{\varrho}$ from Section \ref{['sec:variation']} (blue depicts $|\widetilde{H}(k^{}_{\mathrm{int}})|^2$ and yellow $|H(k^{}_{\mathrm{int}})|^2$). The intensity of the central peak is the same in both cases and given by the point set density, $|\widetilde{H}(0)|^2 = |H(0)|^2 = \mathop{\mathrm{dens}}\nolimits(\varLambda)^2 = \tfrac{\lambda^2}{8} \approx 0.72855\dots$. For the approximation of $|\widetilde{H}(k^{}_{\mathrm{int}})|^2$, the cocycle was used with $n=20$; see Section \ref{['sec:variation']} for further details.
• Figure 3: Diffraction intensities for model sets arising from the substitutions $\varrho$, and $\widetilde{\varrho}$ from Section \ref{['sec:variation']} with weighting (blue for $|\widetilde{H}(k^{}_{\mathrm{int}})|^2$ and yellow for $|H(k^{}_{\mathrm{int}})|^2$) for weighted sets $\widetilde{\varLambda}$ and $\varLambda$ with weights $\alpha = \sqrt{2}$ and $\beta = -1$. The weights are chosen so that the intensity of the central peak vanishes, so $|\widetilde{H}(0)|^2 = |H(0)|^2 = 0$. For the approximation of $|\widetilde{H}(k^{}_{\mathrm{int}})|^2$, the cocycle was again used with $n=20$.
• Figure 4: An illustration of the reprojection technique. While the left figure shows the initial cut-and-project scheme with two orthogonal projections $\pi$, $\pi^{}_{\mathrm{int}}$, the right shows the change of the first projection. The strip $(\mathbb{R}\space\times W) \cap \mathcal{L}$ (shaded area) in the underlying space remains unchanged, but the resulting projection sets are different.
• Figure 5: Diffraction of the deformed model sets $\varLambda'$ (yellow/top) and $\widetilde{\varLambda}'$ (bottom/blue) with weights $\alpha = \sqrt{2}$ and $\beta = -1$. They are chosen so that the central peak vanishes. The figure shows the intensities at all points $k = \tfrac{\sqrt{2}}{4}(m+n\sqrt{2}\,)$ with $m,n\in \{-250,\,\dots,\, 250\}$ in three fundamental domains of the dual lattice $\tfrac{2+\sqrt{2}}{4}\mathbb{Z}$.

Theorems & Definitions (7)

• Proposition 3.1: BGS
• Lemma 3.2
• proof
• Theorem 3.3
• Theorem 3.4
• Proposition 4.1: BGS2
• Theorem 4.2