Introduction to Periodic Geometry and Topology

Anosova, Olga; Kurlin, Vitaliy

doi:10.48550/arxiv.2103.02749

Cited by 4 publications

(4 citation statements)

References 26 publications

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“…We now have a hierarchy of continuous isometry invariants from simple and ultra-fast 10,11,39 to slower but provably complete invariants. [40][41][42] Inverse design aims to make complete invariants invertible, so that the space of all materials can be explored by trying new invariant values, which all give rise to 3D periodic structureshaving found areas with low density (and thus high novelty), the proposed structure can be reconstructed directly from its invariant. The ultimate goal is then to describe a much smaller subspace of invariants whose crystals can be physically synthesized.…”

Section: Conclusion and Discussion Of Novelty And Future Workmentioning

confidence: 99%

Continuous chiral distances for two‐dimensional lattices

2023

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Chirality was traditionally considered a binary property of periodic lattices and crystals. However, the classes of two‐dimensional lattices modulo rigid motion form a continuous space, which was recently parametrized by three geographic‐style coordinates. The four non‐oblique Bravais classes of two‐dimensional lattices form low‐dimensional singular subspaces in the full continuous space. Now, the deviations of a lattice from its higher symmetry neighbors can be continuously quantified by real‐valued distances satisfying metric axioms. This article analyzes these and newer G‐chiral distances for millions of two‐dimensional lattices that are extracted from thousands of available two‐dimensional materials and real crystal structures in the Cambridge Structural Database.

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Section: Conclusion and Discussion Of Novelty And Future Workmentioning

confidence: 99%

Continuous chiral distances for two‐dimensional lattices

2023

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“…The earlier work has studied the following important cases of Problem 1.1: 1-periodic discrete series [5,6,40], 2D lattices [10,42], 3D lattices [9,39,41,47], periodic point sets in R 3 [25,57] and in higher dimensions [2][3][4].…”

Section: Related Work On Point Cloud Classificationsmentioning

confidence: 99%

The strength of a simplex is the key to a continuous isometry classification of Euclidean clouds of unlabelled points

Kurlin¹

2023

Preprint

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This paper solves the continuous classification problem for finite clouds of unlabelled points under Euclidean isometry. The Lipschitz continuity of required invariants in a suitable metric under perturbations of points is motivated by the inevitable noise in measurements of real objects.The best solved case of this isometry classification is known as the SSS theorem in school geometry saying that any triangle up to congruence (isometry in the plane) has a continuous complete invariant of three side lengths.However, there is no easy extension of the SSS theorem even to four points in the plane partially due to a 4parameter family of 4-point clouds that have the same six pairwise distances. The computational time of most past metrics that are invariant under isometry was exponential in the size of the input. The final obstacle was the discontinuity of previous invariants at singular configurations, for example, when a triangle degenerates to a straight line.All the challenges above are now resolved by the Simplexwise Centred Distributions that combine inter-point distances of a given cloud with the new strength of a simplex that finally guarantees the Lipschitz continuity. The computational times of new invariants and metrics are polynomial in the number of points for a fixed Euclidean dimension.

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“…The minimisation over infinitely many rotations was implemented in [21] by sampling and gave approximate algorithms for these metrics. The complete invariant isoset [7] for periodic point sets in R n has a continuous metric that can be approximated [6] with a factor O(n). The metric on invariant density functions [15] required a minimisation over R, so far without approximation guarantees.…”

Section: Definition 23 (Voronoi Domain V (λ))mentioning

confidence: 99%

A complete isometry classification of 3-dimensional lattices

Kurlin¹

2022

Preprint

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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.

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Introduction to Periodic Geometry and Topology

Cited by 4 publications

References 26 publications

Continuous chiral distances for two‐dimensional lattices

Continuous chiral distances for two‐dimensional lattices

The strength of a simplex is the key to a continuous isometry classification of Euclidean clouds of unlabelled points

A complete isometry classification of 3-dimensional lattices

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