Mathematics of 2-dimensional lattices



Kurlin, Vitaliy ORCID: 0000-0001-5328-5351
(2022) Mathematics of 2-dimensional lattices. [Preprint]

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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.

Item Type: Preprint
Additional Information: 51 pages, 20 figures. This version has the full appendix. The new continuous analogues of binary chirality were renamed as G-chiral distances to avoid potential confusion with the classical concept. The latest version is at http://kurlin.org/projects/periodic-geometry-topology/lattices2Dmaths.pdf
Uncontrolled Keywords: math.MG, math.MG, 52C07, 52C25, 51N20, 51K05
Depositing User: Symplectic Admin
Date Deposited: 29 Nov 2022 15:48
Last Modified: 14 Mar 2024 17:32
DOI: 10.48550/arxiv.2201.05150
Related URLs:
URI: https://livrepository.liverpool.ac.uk/id/eprint/3166414