Tetrahedron

Combinatorics

Physics

Euclidean geometry

Congruence (geometry)

Integer lattice

Euclidean space

Metric space

Pure mathematics

Acoustics

Geometry

Lattice (music)

Mathematics

Half-integer

Quantum mechanics

Isometry (Riemannian geometry)

Scaling

Foundations of Computational Mathematics

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 partially resolved, but standard reductions remain discontinuous under perturbations modelling atomic displacements. 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 lattice deviations from higher-symmetry neighbours. The geometric methods extend the past work of Delone, Conway, and Sloane.

Mathematics of 2-Dimensional Lattices