Geographic style maps for two-dimensional lattices

Bright, Matthew; Cooper, Andrew I.; Kurlin, Vitaliy

doi:10.1107/s2053273322010075

Cited by 11 publications

(12 citation statements)

References 32 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…fold intersection [− 1 12 ,7 12 ] of the lengthψ 3 ( 1 2 ) = 2 3, which coincides with ψ 0 (0) = 2 3 .…”

mentioning

confidence: 71%

See 1 more Smart Citation

Density functions of periodic sequences of continuous events

Anosova¹,

Kurlin²

2023

Preprint

View full text Add to dashboard Cite

Periodic Geometry studies isometry invariants of periodic point sets that are also continuous under perturbations. The motivations come from periodic crystals whose structures are determined in a rigid form but any minimal cells can discontinuously change due to small noise in measurements. For any integer k ≥ 0, the density function of a periodic set S was previously defined as the fractional volume of all k-fold intersections (within a minimal cell) of balls that have a variable radius t and centers at all points of S. This paper introduces the density functions for periodic sets of points with different initial radii motivated by atomic radii of chemical elements and by continuous events occupying disjoint intervals in time series. The contributions are explicit descriptions of the densities for periodic sequences of intervals. The new densities are strictly stronger and distinguish periodic sequences that have identical densities in the case of zero radii.

show abstract

“…fold intersection [− 1 12 ,7 12 ] of the lengthψ 3 ( 1 2 ) = 2 3, which coincides with ψ 0 (0) = 2 3 .…”

mentioning

confidence: 71%

“…The subarea of Lattice Geometry developed continuous parameterizations for the moduli spaces of lattices considered up to isometry in dimension two [7,13] and three [6,10].…”

Section: Review Of Related Past Workmentioning

confidence: 99%

Density functions of periodic sequences of continuous events

Anosova¹,

Kurlin²

2023

Preprint

View full text Add to dashboard Cite

show abstract

“…Figure 9 (center) shows that if we plot the density of 2D lattices generated from the CSD in the QT, we observe a strong preference for higher symmetry structures (on the boundary) and a concentration of lattices toward the point (1, 0) representing the hexagonal lattice, while density decreases sharply toward the point (0, 1) representing infinitely long, thin lattices. 15…”

Section: Results: G-chiral Distances For 2d Lattices Of 3d Crystalsmentioning

confidence: 99%

“…This article completes the non-trivial case of twodimensional lattices 14 and uses geographical-style mapping 15 to define continuous chiral distances in a way that is not based on discrete symmetry groups of the lattice. A point in the Euclidean plane ℝ 2 is identified with a vector v = (x, y), whose length is j v j¼ ffiffiffiffiffiffiffiffiffiffiffiffiffi ffi…”

Section: Introduction: Continuous G-chiral Distances Of Latticesmentioning

confidence: 97%

Continuous chiral distances for two‐dimensional lattices

2023

Self Cite

View full text Add to dashboard Cite

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.

show abstract

“…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

View full text Add to dashboard Cite

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.

show abstract

Geographic style maps for two-dimensional lattices

Cited by 11 publications

References 32 publications

Density functions of periodic sequences of continuous events

Density functions of periodic sequences of continuous events

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

Contact Info

Product

Resources

About