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

TL;DR

This paper explicitly derives and computes the diffraction spectra of the Hat and Spectre tilings, revealing their pure point nature through model set representations and fractal boundary analysis.

Contribution

It introduces a method to compute diffraction spectra of complex tilings using model sets and renormalisation techniques, specifically for Hat and Spectre tilings.

Findings

Diffraction spectra are pure point for Hat and Spectre tilings.

Explicit Fourier--Bohr coefficients are calculated.

Model set representations facilitate analysis of fractal boundary effects.

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.