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 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 past work of Delone, Conway and Sloane.1 The hard problem to continuously classify lattices up to isometry We extend the continuous isometry classification of 2-dimensional lattices [19] to dimension 3. A lattice Λ ⊂ R n consists of integer linear combinations of basis vectors v 1 , . . . , v n . This basis spans a parallelepiped called a unit cell U ⊂ R n .The problem to classify lattices up to isometry is motivated by periodic crystals whose structures are determined in a rigid form. Hence the most natural equivalence of crystals is rigid motion. We start from general isometries that also include mirror reflections because the sign of a lattice similar to [19, Definition 3.4] easily distinguishes mirror images. As in R 2 , the space of lattices up to rigid motion in R 3 is a 2-fold cover of the smaller Lattice Isometry Space LIS(R 3 ).The previous work [19, section 1] provided important motivations for a continuous classification problem, which we state below for 3-dimensional lattices.

Geographic style maps for two-dimensional lattices

“…Kurlin (2022b, lemma 3.8) proved that RI(Ã) is an isometry invariant of Ã, independent of an obtuse superbase B because an obtuse superbase of à is unique up to isometry, also up to rigid motion for non-rectangular lattices. This uniqueness was missed by Conway & Sloane (1992) and actually fails in R 3 (see Kurlin, 2022a). Left: the triangular cone TC = {ðr 12 ; r 01 ; r 02 Þ 2 R 3 j 0 r 12 r 01 r 02 6 ¼ 0} is the space of all root invariants, see Definition 3.1.…”

“…Though the paper by Conway & Sloane (1992) 30 years ago aimed for continuous invariants of three-dimensional lattices, no formal proofs were given even for the isometry invariance. This past work for three-dimensional lattices has been corrected and extended by Kurlin (2022a). Kurlin (2022b, proposition 3.10) proves that a reduced basis from Definition 2.1 is unique (also in the case of rigid motion) and all reduced bases are in a 1-1 correspondence with obtuse superbases, which are easier to visualize, especially for n 3.…”

An invertible seven-dimensional Dirichlet cell characterization of lattices

“…Reversing Allman's observation that a Buerger-reduced cell is a good stepping stone to a Selling-reduced cell (Allmann, 1968), a Selling-reduced cell can be an efficient stepping stone to a Buerger-reduced cell that quickly reduces to a Niggli-reduced cell (Andrews et al, 2019). Note that the negatives of the Selling scalars of a Selling-reduced cell are non-negative so that the six square roots provide a convenient six-parameter characterization of a lattice (Kurlin, 2022).…”