Square-triangle tilings: an infinite playground for soft matter

Abstract

Regular square and triangle, two very simple geometrical figures, can be used to construct a fascinating variety of tilings which cover the 2D plane without any overlaps or holes. Such tilings are observed in many soft matter systems. Here we present a way to describe all possible globally uniform square-triangle phases using a three dimensional composition space. This approach takes into account both the overall composition and the orientations of the two kinds of tiles. The geometrical properties of special phases encountered in soft matter systems are described: the Archimedean Σ and H phases, the striped phases and the 12-fold maximally symmetric phases. We show how this very rich behavior with either periodic or aperiodic phases appears here as a consequence of the inherent incommensurability between the areas of the two tiles related by the ratio Image ID:d1sm01242h-t1.gif. Geometrical constraints on boundary lines and junction points between domains of different compositions are given, a situation likely to be encountered in experimental and numerical studies. Future developments are suggested like considering the effect on phase behavior of possible symmetry breaking.

Graphical abstract: Square-triangle tilings: an infinite playground for soft matter

Supplementary files

Article information

Article type
Paper
Submitted
26 Aug 2021
Accepted
28 Sep 2021
First published
13 Oct 2021

Soft Matter, 2021, Advance Article

Square-triangle tilings: an infinite playground for soft matter

M. Impéror-Clerc, A. Jagannathan, P. Kalugin and J. Sadoc, Soft Matter, 2021, Advance Article , DOI: 10.1039/D1SM01242H

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