Symmetries of (quasi)periodic materials: Superposability vs. Indistinguishability

TL;DR

This paper investigates the symmetries of (quasi)periodic materials using a statistical approach based on autocorrelation functions and Fourier analysis, introducing an image processing method for symmetry identification.

Contribution

It presents a novel methodology combining autocorrelation and Fourier analysis to determine space groups of (quasi)periodic materials from images.

Findings

Method successfully identifies space group characteristics from synthetic images.

The Fourier-based approach reveals the rotational symmetry order of Penrose tiling as ten.

Validation on synthetic data demonstrates effectiveness of the proposed technique.

Abstract

This work is devoted to the study of the symmetries of (quasi)periodic architectured materials. For this purpose, the weaker symmetry criterion of indistinguishability is used. It relies on a statistical description of the mesostructure and is defined in terms of the spatial autocorrelation functions of the material under consideration. By using the representation of these autocorrelation functions in Fourier space, the space groups of both periodic and quasiperiodic materials can be obtained. In this context, an image processing methodology is proposed to identify the key characteristics of a material's space group (i.e its point group and its symmorphism) directly from the Fourier transform of the mesostructure. The method is validated on synthetic two-dimensional images of (quasi)periodic architectured materials and it is pointed out, as an illustrative example, that the rotational symmetry of the classical Penrose tiling is of order ten.