Project Periodic Geometry within a new area of Geometric Data Science based on papers in
CGD 2024
IUCrJ 2024
SISC 2023
NeurIPS 2022
PRE 2022
MATCH 2022
DGMM 2021
 Periodic Geometry develops continuous parametrizations for moduli spaces of periodic point sets up to isometry.
 The subarea called Lattice Geometry studies moduli spaces of simpler periodic lattices in low dimensions 2 and 3.
 The subarea called Density Functions studies generically complete isometry invariants extending the crystal density.
 The adjacent area of Cloud Isometry Spaces studies geometry of moduli spaces of finite clouds of unordered points.
 The related area of StructureProperty Relations aims to explain all properties of a structure in terms of its invariants.
 The applied area of Computational Materials Science explores practical applications of geometric invariants and metrics.
 The wider area of Geometric Data Science studies moduli spaces of any data objects up to practical equivalences.
 The latest developments are discussed in the MIF++ seminar and at the annual conference MACSMIN since 2020.
Continuous invariant maps of a crystal database

 Abstract. The Cambridge Structural Database (CSD) played a key role in the recently established Crystal Isometry Principle (CRISP). The CRISP says that any real periodic crystal is uniquely determined as a rigid structure by the geometry of its atomic centres without atomic types. Ignoring atomic types allows us to study all periodic crystals in a common space whose continuous nature is justified by the continuity of realvalued coordinates of atoms. % real values, not restricted to a discrete grid. Our previous work introduced structural descriptors PDD (Pointwise Distance Distributions) that are invariant under isometry defined as a composition of translations, rotations, and reflections. The PDD invariants distinguished all nonduplicate periodic crystals in the CSD. This paper presents the first continuous maps of the CSD and its important subsets in invariant coordinates that have analytic formulae and physical interpretations. Any existing periodic crystal has a uniquely defined location on these geographicstyle maps. Any newly discovered periodic crystals will appear on the same maps without disturbing the past materials.
@article{widdowson2024continuous, author = {Daniel Widdowson and Vitaliy Kurlin}, title = {Continuous invariantbased maps of the Cambridge Structural Database}, journal = {Crystal Growth and Design}, volume = {24}, issue = {13}, pages = {5627–5636}, year = {2024}, doi = {10.1021/acs.cgd.4c00410} }
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Important definitions in crystallography

 Abstract. This paper was motivated by the articles ``Same or different  that is the question'' in CrystEngComm (July 2020) and ``Change to the definition of a crystal'' in the IUCr newsletter (June 2021). Experimental approaches to crystal comparisons require rigorously defined classifications in crystallography and beyond. Since crystal structures are determined in a rigid form, their strongest equivalence in practice is rigid motion, which is a composition of translations and rotations in 3dimensional space. Conventional representations based on reduced cells and standardizations theoretically distinguish all periodic crystals. However, all cellbased representations are inherently discontinuous under almost any atomic displacement that can arbitrarily scale up a reduced cell. Hence comparing millions of known structures in materials databases needs continuous distance metrics.
@article{anosova2024importance, author = {Olga Anosova and Vitaliy Kurlin and Marjorie Senechal}, title = {The importance of definitions in crystallography}, journal = {IUCrJ}, volume = {11}, issue = {4}, pages = {453463}, doi = {10.1107/S2052252524004056}, year = {2024} }
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Densest Crystallographic Group Packings

 Abstract. Molecular crystal structure prediction (CSP) seeks the most stable periodic structure given the chemical composition of molecule and pressuretemperature conditions. Modern CSP solvers use global optimization methods to search for structures with minimal free energy within a complex energy landscape induced by intermolecular potentials. A major caveat of these methods is that initial configurations are random, making the search susceptible to convergence at local minima. Providing initial configurations that are densely packed with respect to the geometric representation of a molecule can significantly accelerate CSP. Motivated by these observations, we define a class of periodic packings restricted to crystallographic symmetry groups (CSG) and design a search method for the densest CSG packings in an informationgeometric framework. Since CSG induces a toroidal topology on the configuration space, a nonEuclidean trust region method is performed on a statistical manifold consisting of probability distributions defined on an ndimensional flat unit torus by extending the multivariate von Mises distribution. Introducing an adaptive quantile reformulation of the fitness function into the optimization schedule provides the algorithm with a geometric characterization through local dual geodesic flows. Moreover, we examine the geometry of the adaptive selectionquantile defined trust region and show that the algorithm performs a maximization of stochastic dependence among elements of the extended multivariate von Mises distributed random vector. We experimentally evaluate the behavior and performance of the algorithm on various densest packings of convex polygons in 2dimensional CSGs for which optimal solutions are known, and we demonstrate its application in the pentacene thinfilm CSP.
@article{torda2023entropic, author = {Milo Torda and John Goulermas and Roland Púček and Vitaliy Kurlin}, title = {Entropic Trust Region for Densest Crystallographic Symmetry Group Packings}, journal = {SIAM Journal on Scientific Computing}, volume = {45}, number = {4}, pages = {B493B522}, year = {2023}, doi = {10.1137/22M147983X} }
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Resolving the data ambiguity for periodic crystals

 Abstract. The fundamental model of all solid crystalline materials (periodic crystals) is a periodic set of atomic centers considered up to rigid motion in Euclidean space. The major obstacle to materials discovery was highly ambiguous representations that didn't allow fast and reliable comparisons, and led to numerous (near) duplicates in all experimental databases. This paper introduces the new invariants that are crystal descriptors without false negatives and are called Pointwise Distance Distributions (PDD). The PDD invariants are numerical matrices with a nearlinear time complexity and an exactly computable metric. The strongest theoretical result is generic completeness (absence of false positives) for all finite and periodic sets of points in any dimension. The strength of PDD is demonstrated by 200B+ pairwise comparisons of all 660K+ periodic structures from the world's largest Cambridge Structural Database of 1.17M+ known crystals over two days on a modest desktop.
@article{widdowson2022resolving, author = {Daniel Widdowson and Vitaliy Kurlin}, title = {Resolving the data ambiguity for periodic crystals}, journal = {Advances in Neural Information Processing Systems (Proceedings of NeurIPS 2022)}, volume = {35}, pages = {2462524638}, year = {2022} }
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Densest packings of regular polygons

 DOI : 10.1103/PhysRevE.106.054603
 Abstract. Packings of regular convex polygons (ngons) that are sufficiently dense have been studied extensively in the context of modeling physical and biological systems as well as discrete and computational geometry. Former results were mainly regarding densest lattice or doublelattice configurations. Here we consider all 2dimensional crystallographic symmetry groups (plane groups) by restricting the configuration space of the general packing problem of congruent copies of a compact subset of the 2dimensional Euclidean space to particular isomorphism classes of the discrete group of isometries. We formulate the plane group packing problem as a nonlinear constrained optimization problem. By means of the Entropic Trust Region Packing Algorithm that approximately solves this problem, we examine some known and unknown densest packings of various ngons in all 17 plane groups and state conjectures about common symmetries of the densest plane group packings for every ngon.
@article{torda2022densest, author = {Miloslav Torda and John Y Goulermas and Vitaliy A Kurlin and Graeme M Day}, title = {Densest plane group packings of regular polygons}, journal = {Physical Review E}, volume = {106}, issue = {5}, pages = {054603}, year = {2022} }
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Average Minimum Distances of periodic point sets

 DOI : 10.46793/match.873.529W
 Abstract. The fundamental model of any solid crystalline material (crystal) at the atomic scale is a periodic point set. The strongest natural equivalence of crystals is rigid motion or isometry that preserves all interatomic distances. Past comparisons of periodic structures often used manual thresholds, symmetry groups and reduced cells, which are discontinuous under perturbations or thermal vibrations of atoms. This work defines the infinite sequence of continuous isometry invariants (Average Minimum Distances) to progressively capture distances between neighbours. The asymptotic behaviour of the new invariants is theoretically proved in all dimensions for a wide class of sets including nonperiodic. The proposed near linear time algorithm identified all different crystals in the world's largest Cambridge Structural Database over a few hours on a modest desktop. The computational strength provides rigorous foundations to continuously parametrise the space of all real periodic crystals as a highdimensional extension of Mendeleev's periodic table of elements.
@article{widdowson2022average, author = {Daniel Widdowson and Marco Mosca and Angeles Pulido and Andrew Cooper and Vitaliy Kurlin}, title = {Average Minimum Distances of periodic point sets  foundational invariants for mapping all periodic crystals}, journal = {MATCH Communications in Mathematical and in Computer Chemistry}, volume = {87}, issue = {3}, pages = {529559}, year = {2022} }
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Crystal isosets are complete isometry invariants

 DOI : 10.1007/9783030766573_16
 Abstract. We develop discrete geometry methods to resolve the data ambiguity challenge for periodic point sets to accelerate materials discovery. In any highdimensional Euclidean space, a periodic point set is obtained from a finite set (motif) of points in a parallelepiped (unit cell) by periodic translations of the motif along basis vectors of the cell. An important equivalence of periodic sets is a rigid motion or an isometry that preserves interpoint distances. This equivalence is motivated by solid crystals whose periodic structures are determined in a rigid form. Crystals are still compared by descriptors that are either not isometry invariants or depend on manually chosen tolerances or cutoff parameters. All discrete invariants including symmetry groups can easily break down under atomic vibrations, which are always present in real crystals. We introduce a complete isometry invariant for all periodic sets of points, which can additionally carry labels such as chemical elements. The main classification theorem says that any two periodic sets are isometric if and only if their proposed complete invariants (called isosets) are equal. A potential equality between isosets can be checked by an algorithm, whose computational complexity is polynomial in the number of motif points. The key advantage of isosets is continuity under perturbations, which allows us to quantify similarities between any periodic sets.
@inproceedings{anosova2021isometry, author = {Anosova, Olga and Kurlin, Vitaliy}, title = {An isometry classification of periodic point sets}, booktitle = {Lecture Notes in Computer Science (Proceedings of DGMM)}, volume = {12708}, pages = {229241}, year = {2021} }
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