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Долбилин, Николай Петрович; Bouniaev, Mikhail

doi:10.1016/j.ejc.2018.02.005

Cited by 5 publications

(7 citation statements)

References 1 publication

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“…The latest invariant isoset [4] reduces the isometry classification of all periodic point sets to a fintie collection of isometry classes of α-clusters around points in a motif at a certain radius α, which was motivated by the seminal work of Dolbilin with co-authors about Delone sets [7,15,16]. Checking if two isosets coincide needs a cubic algorithm, which is not yet implemented.…”

Section: Periodic Point Sets and A Review Of Past Workmentioning

confidence: 99%

Average minimum distances of periodic point sets – foundational invariants for mapping periodic crystals

Widdowson¹,

Mosca²,

Pulido³

et al. 2021

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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 inter-atomic 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 neighbors. The asymptotic behaviour of the new invariants is theoretically proved in all dimensions for a wide class of sets including non-periodic. The proposed near linear time algorithm identified all different crystals in the world's largest Cambridge Structural Database within a few hours on a modest desktop. The ultra fast speed and proved continuity provide rigorous foundations to continuously parameterise the space of all periodic crystals as a high-dimensional extension of Mendeleev's table of elements.

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Section: Periodic Point Sets and A Review Of Past Workmentioning

confidence: 99%

Average minimum distances of periodic point sets – foundational invariants for mapping periodic crystals

Widdowson¹,

Mosca²,

Pulido³

et al. 2021

match

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“…More recently, for any periodic point set S ⊂ R n with a motif M in a unit cell U , Edelsbrunner et al [18] The latest advance [5] reduces the isometry classification of all periodic point sets to an isoset of isometry classes of α-clusters around points in a motif at a certain radius α, which was motivated by the seminal work of Dolbilin with co-authors about Delone sets [16,8,15].…”

Section: Definitions and A Review Of Past Work On Comparing Periodic ...mentioning

confidence: 99%

Pointwise distance distributions of periodic point sets

Widdowson¹,

Kurlin²

2021

Preprint

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The fundamental model of a periodic structure is a periodic set of points considered up to rigid motion or isometry in Euclidean space. The recent work by Edelsbrunner et al defined the new isometry invariants (density functions), which are continuous under perturbations of points and complete for generic sets in dimension 3. This work introduces much faster invariants called higher order Pointwise Distance Distributions (PDD). The new PDD invariants are simpler represented by numerical matrices and are also continuous under perturbations important for applications.Completeness of PDD invariants is proved for distance-generic sets in any dimension, which was also confirmed by distinguishing all 229K known molecular organic structures from the world's largest Cambridge Structural Database. This huge experiment took only seven hours on a modest desktop due to the proposed algorithm with a near linear or small polynomial complexity in terms of key input sizes. Most importantly, the above completeness allows one to build a common map of all periodic structures, which are continuously parameterized by PDD and explicitly reconstructible from PDD. Appendices include first tree-based maps for several thousands of real structures.

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“…The key novelty of Periodic Geometry is the aim to study the continuous space of all isometry classes of periodic structures, not isolated structures as in the past. Related results from [14,8,12] and [19,24] will be reviewed in sections 6 and 9.…”

Section: Definition 4 (Delone Sets)mentioning

confidence: 99%

Introduction to Periodic Geometry and Topology

Anosova¹,

Kurlin²

2021

Preprint

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This paper introduces the key concepts and problems of the new research area of Periodic Geometry and Topology for applications in Materials Science. Periodic structures such as solid crystalline materials or textiles were previously studied as isolated structures without taking into account the continuity of their configuration spaces. The key new problem in Periodic Geometry is an isometry classification of periodic point sets. A required complete invariant should continuously change under point perturbations, because atoms always vibrate in real crystals. The main objects of Periodic Topology are embeddings of curves in a thickened plane that are invariant under lattice translations. Such periodic knots were classified in the past up to continuous deformations (isotopies) that keep a fixed lattice structure, hence are realized in a fixed thickened torus. The more practical equivalence is a periodic isotopy in a thickened plane without fixing a lattice basis. The paper states the first results in the new area and proposes further problems and directions.

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Regular t-bonded systems in R3

Cited by 5 publications

References 1 publication

Average minimum distances of periodic point sets – foundational invariants for mapping periodic crystals

Average minimum distances of periodic point sets – foundational invariants for mapping periodic crystals

Pointwise distance distributions of periodic point sets

Introduction to Periodic Geometry and Topology

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