Geographic-style maps for 2-dimensional lattices

TL;DR

This paper introduces geographic-style maps for 2D crystal lattices, visualizing all known crystal classes and thousands of real lattices, aiding in the development of invariant-based crystallography.

Contribution

It develops continuous, invariant-based maps of 2D lattices, visualizing all crystal classes and real crystal data in a unified geometric framework.

Findings

All Bravais classes are represented as low-dimensional subspaces.

Visualizations include hundreds of thousands of real crystal lattices.

The maps facilitate the development of continuous, invariant-based crystallography.

Abstract

This paper develops geographic-style maps containing 2D lattices in all known crystals parameterised by recent complete invariants. Motivated by rigid crystal structures, lattices are considered up to rigid motion and uniform scaling. The resulting space of 2D lattices is a square with identified edges or a sphere without one point. The new continuous maps show all Bravais classes as low-dimensional subspaces, visualise hundreds of thousands of real crystal lattices from the Cambridge Structural Database, and motivate the development of continuous and invariant-based crystallography.