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

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.

Figures (29)

• Figure 1: Characteristic scales of an architectured material
• Figure 2: Laser cut disk of Penrose honeycomb used for Brazilian test somera2022effective.
• Figure 3: Illustration of the mismatch of the Penrose tiling for a $\frac{2\pi}{10}$ rotation combined with a translation. On figure (a) two five-pointed stars with different orientations are highlighted in yellow and blue. On figure (b) defect lines (worms) arise from the lack of translational invariance in the tiling.
• Figure 4: Square lattice group
• Figure 5: Hexagonal lattice group