Some comments on pinwheel tilings and their diffraction

TL;DR

This paper explores the diffraction pattern of pinwheel tilings, highlighting their circular symmetry and analyzing their autocorrelation and diffraction measures through combinatorial and numerical methods.

Contribution

It provides new insights into the diffraction properties of pinwheel tilings, including their circular symmetry and the impact of tile orientations.

Findings

Diffraction consists of sharp rings and possibly a continuous component.

Autocorrelation exhibits circular symmetry.

Numerical methods help analyze weighted pinwheel point sets.

Abstract

The pinwheel tiling is the paradigm for a substitution tiling with circular symmetry, in the sense that the corresponding autocorrelation is circularly symmetric. As a consequence, its diffraction measure is also circularly symmetric, so the pinwheel diffraction consists of sharp rings and, possibly, a continuous component with circular symmetry. We consider some combinatorial properties of the tiles and their orientations, and a numerical approach to the diffraction of weighted pinwheel point sets.