Hexagonal Projected Symmetries

TL;DR

This paper classifies 3D crystallographic groups whose planar projections exhibit hexagonal symmetry, aiding the understanding of pattern formation in thin physical domains with complex symmetries.

Contribution

It provides a constructive classification of crystallographic groups leading to hexagonal projected symmetries, expanding the understanding of pattern formation in symmetric systems.

Findings

Identifies specific crystallographic groups with hexagonal projected symmetries

Provides a constructive method for determining projected symmetries

Enables analysis of complex patterns like the black-eye pattern in thin domains

Abstract

In the study of pattern formation in symmetric physical systems a 3-dimensional structure in thin domains is often modelled as 2-dimensional one. We are concerned with functions in $R^3$ that are invariant under the action of a crystallographic group and the symmetries of their projections into a function defined on a plane. We obtain a list of the crystallographic groups for which the projected functions have a hexagonal lattice of periods. The proof is constructive and the result may be used in the study of observed patterns in thin domains, whose symmetries are not expected in 2-dimensional models, like the black-eye pattern.