A complete isometry classification of 3-dimensional lattices

Abstract

A periodic lattice in Euclidean 3-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 3-dimensional lattices up to Euclidean isometry (or congruence) and similarity (with uniform scaling).The new homogeneous invariants are uniquely ordered square roots of scalar products of four superbase vectors whose sum is zero and all pairwise angles are non-acute. These root invariants continuously change under perturbations of basis vectors. The geometric methods extend the work of Delone, Conway and Sloane.