Mathematics of 2-dimensional lattices

TL;DR

This paper provides a complete continuous classification and new invariants for 2D lattices, enabling easy computation of metrics and continuous measurement of lattice deviations from symmetry.

Contribution

It introduces homogeneous invariants that allow a continuous classification of 2D lattices up to isometry, rigid motion, and similarity, resolving previous discontinuity issues.

Findings

Established continuous invariants for 2D lattices

Developed metrics for lattice comparison under various equivalences

Introduced chiral distances measuring lattice symmetry deviations

Abstract

A periodic lattice in Euclidean space is the infinite set of all integer linear combinations of basis vectors. Any lattice can be generated by infinitely many different bases. This ambiguity was only partially resolved, but standard reductions remained discontinuous under perturbations modelling crystal vibrations. This paper completes a continuous classification of 2-dimensional lattices up to Euclidean isometry (or congruence), rigid motion (without reflections), and similarity (with uniform scaling). The new homogeneous invariants allow easily computable metrics on lattices considered up to the equivalences above. The metrics up to rigid motion are especially non-trivial and settle all remaining questions on (dis)continuity of lattice bases. These metrics lead to real-valued chiral distances that continuously measure a lattice deviation from a higher-symmetry neighbour.