Visual maps of the isometry space of 3-dimensional lattices.

10^th December 2021, 14:00
Vitaly Kurlin
MIF

Abstract

Since crystal structures are determined in a rigid form, 3-dimensional periodic crystals and their lattices should be studied up to rigid motion or isometry, which are compositions of translations, rotations, and reflections. The resulting space of isometry classes of lattices is infinite and was previously studied mostly by discrete invariants, which split this continuous space into a finite number of strata or low-dimensional subspaces representing high-symmetry lattices. We extend the continuously parametrised map of all lattices in dimension 2 (joint with Matt Bright and Andy Cooper) to dimension 3 and also introduce easy metrics on lattices, which are continuous under perturbations of lattice bases.