Lattice Geometry within a new area of Geometric Data Science based on papers in
Chirality 2023
ACA 2023
FoCM 2022
CRaT 2020
CCCG 2015
 Lattice Geometry develops continuous parametrisations and metrics for moduli spaces of lattices up to isometry.
 The wider area of Periodic Geometry studies moduli spaces of general periodic point sets that model all periodic crystals.
 The related area of Cloud Isometry Spaces studies geometry of moduli spaces of finite clouds of unlabeled points.
 The even wider area of Geometric Data Science studies moduli spaces of any data objects up to practical equivalences.
 The applied area of Computational Materials Science explores practical applications of geometric invariants and metrics.
 The latest developments are discussed in the MIF++ seminar and at the annual conference MACSMIN since 2020.
Continuous chiral distances for 2D lattices

 DOI : doi:10.1002/chir.23598
 Abstract. Chirality was traditionally considered a binary property of periodic lattices and crystals. However, the classes of 2dimensional lattices modulo rigid motion form a continuous space, which was recently parametrized by three geographicstyle coordinates. The four nonoblique Bravais classes of 2dimensional lattices form lowdimensional singular subspaces in the full continuous space. Now the deviations of a lattice from its higher symmetry neighbours can be continuously quantified by realvalued distances satisfying metric axioms. This paper analyses these and newer Gchiral distances for millions of 2dimensional lattices that are extracted from publicly available databases of 2dimensional structures and real materials in the Cambridge Structural Database.
@article{bright2023continuous, title={Continuous chiral distances for 2dimensional lattices}, author={Matthew J Bright and Andrew I Cooper and Vitaliy A Kurlin}, journal={Chirality}, doi={10.1002/chir.23598}, year={2023} }
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Geographicstyle maps for 2D lattices

 DOI : 10.1107/S2053273322010075
 Abstract. This paper develops geographicstyle maps containing 2D lattices in all known periodic crystals parameterised by recent complete invariants. Motivated by rigid crystal structures, lattices are considered up to rigid motion and uniform scaling. The resulting space of 2D lattices is a square with identified edges or a punctured sphere. The new continuous maps show all Bravais classes as lowdimensional subspaces, visualise hundreds of thousands of real crystal lattices from the Cambridge Structural Database, and motivate the development of continuous and invariantbased crystallography.
@article{bright2023geographic, title={Geographicstyle maps for 2dimensional lattices}, author={Matthew J Bright and Andrew I Cooper and Vitaliy A Kurlin}, journal={Acta Crystallographica Section A}, volume ={79}, number ={1}, pages={113}, year={2023} }
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Mathematics of 2D lattices

 DOI : 10.1007/s10208022096018
 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 remain discontinuous under perturbations modelling crystal vibrations. This paper completes a continuous classification of 2dimensional 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 nontrivial and settle all remaining questions on (dis)continuity of lattice bases. These metrics lead to realvalued chiral distances that continuously measure lattice deviations from highersymmetry neighbours. The geometric methods extend the work of Delone, Conway, and Sloane.
@article{kurlin2022mathematics, title={Mathematics of 2dimensional lattices}, author={Vitaliy A Kurlin}, journal={Foundations of Computational Mathematics}, pages={159}, doi={10.1007/s10208022096018}, year={2022} }
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Voronoibased metrics on lattices

 DOI : 10.1002/crat.201900197
 Abstract. This paper develops a new continuous approach to a similarity between periodic lattices of ideal crystals. Quantifying a similarity between crystal structures is needed to substantially speed up the Crystal Structure Prediction, because the prediction of many target properties of crystal structures is computationally slow and is essentially repeated for many nearly identical simulated structures. The proposed distances between arbitrary periodic lattices of crystal structures are invariant under all rigid motions, satisfy the metric axioms and continuity under atomic perturbations. The above properties make these distances ideal tools for clustering and visualizing large datasets of crystal structures. All the conclusions are rigorously proved and justified by experiments on real and simulated crystal structures reported in the Nature 2017 paper Functional materials discovery using energy–structure–function maps.
@article{mosca2020voronoi, title={Voronoibased similarity distances between arbitrary crystal lattices}, author={Mosca, Marco and Kurlin, Vitaliy}, journal={Crystal Research and Technology}, volume={55}, number={5}, pages={190197}, year={2020} }
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Relaxed disk packing

 Abstract. Motivated by biological questions, we study configurations of equalsized disks in the Euclidean plane that neither pack nor cover. Measuring the quality by the probability that a random point lies in exactly one disk, we show that the regular hexagonal grid gives the maximum among lattice configurations.
@inproceedings{edelsbrunner2015relaxed, author = {Herbert Edelsbrunner and Mabel IglesiasHam and Vitaliy A Kurlin}, title = {Relaxed disk packing}, booktitle = {Proceedings of CCCG 2015: Canadian Conference on Computational Geometry}, year = {2015} }
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