Pinwheel patterns and powder diffraction

TL;DR

This paper explores the unique symmetries of pinwheel patterns and their diffraction images, developing a framework to analyze their statistical properties and comparing them with models like powder diffraction of square lattices.

Contribution

It introduces a new approach using an alternative substitution rule to study the diffraction properties of pinwheel patterns and their higher-dimensional generalizations.

Findings

Pinwheel patterns exhibit continuous circular symmetries in diffraction images.

The statistical properties of these patterns can be analyzed using a novel substitution rule.

Similarities are found between pinwheel diffraction and powder diffraction of regular crystals.

Abstract

Pinwheel patterns and their higher dimensional generalisations display continuous circular or spherical symmetries in spite of being perfectly ordered. The same symmetries show up in the corresponding diffraction images. Interestingly, they also arise from amorphous systems, and also from regular crystals when investigated by powder diffraction. We present first steps and results towards a general frame to investigate such systems, with emphasis on statistical properties that are helpful to understand and compare the diffraction images. We concentrate on properties that are accessible via an alternative substitution rule for the pinwheel tiling, based on two different prototiles. Due to striking similarities, we compare our results with the toy model for the powder diffraction of the square lattice.