Square-triangle tilings: an infinite playground for soft matter.

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.

Robust minimal matching rules for quasicrystals.

A unified framework is proposed for dealing with matching rules of quasiperiodic patterns, relevant for both tiling models and real-world quasicrystals. The approach is intended for extraction and validation of a minimal set of matching rules, directly from the phased diffraction data. The construction yields precise values for the spatial density of distinct atomic positions and tolerates the presence of defects in a robust way.