Universally optimal distribution of points on spheres

Cohn, Henry; Kumar, Abhinav

doi:10.1090/s0894-0347-06-00546-7

Cited by 329 publications

(480 citation statements)

References 46 publications

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“…They show that the D 4 kissing configuration of 24 points does not form a universally optimal spherical configuration, by finding a competing family of configurations that occasionally beats it. (By contrast, Cohn and Kumar [12] proved that the E 8 kissing configuration is universally optimal.) Unfortunately, the spherical competitors do not seem to extend to Euclidean packings.…”

Section: Conclusion and Discussionmentioning

confidence: 96%

Counterintuitive ground states in soft-core models

Cohn

Kumar

2008

Phys. Rev. E

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It is well known that statistical mechanics systems exhibit subtle behavior in high dimensions. In this paper, we show that certain natural soft-core models, such as the Gaussian core model, have unexpectedly complex ground states even in relatively low dimensions. Specifically, we disprove a conjecture of Torquato and Stillinger, who predicted that dilute ground states of the Gaussian core model in dimensions 2 through 8 would be Bravais lattices. We show that in dimensions 5 and 7, there are in fact lower-energy non-Bravais lattices. (The nearest three-dimensional analog is the hexagonal close-packing, but it has higher energy than the face-centered cubic lattice.) We believe these phenomena are in fact quite widespread, and we relate them to decorrelation in high dimensions.

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Section: Conclusion and Discussionmentioning

confidence: 96%

Counterintuitive ground states in soft-core models

Cohn

Kumar

2008

Phys. Rev. E

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“…The sphere packing problem is equivalent to solving for s ¼ 1. Cohn & Kumar (2007) have shown that there exist configurations for certain values of N which are universally optimal, that is, globally optimal solutions for every value of s. The known universally optimal configurations for d ¼ 3 are the tetrahedron, the 16-cell and the 600-cell. The vertices of these polyhedra are conjectured to be global optima for the sphere covering problem, since their Delaunay triangulations consist of regular spherical tetrahedra (cf.…”

Section: Riesz Energy Minimizationmentioning

confidence: 99%

Improved orientation sampling for indexing diffraction patterns of polycrystalline materials

Larsen¹,

Schmidt²

2017

J Appl Cryst

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Orientation mapping is a widely used technique for revealing the microstructure of a polycrystalline sample. The crystalline orientation at each point in the sample is determined by analysis of the diffraction pattern, a process known as pattern indexing. A recent development in pattern indexing is the use of a bruteforce approach, whereby diffraction patterns are simulated for a large number of crystalline orientations and compared against the experimentally observed diffraction pattern in order to determine the most likely orientation. Whilst this method can robustly identify orientations in the presence of noise, it has very high computational requirements. In this article, the computational burden is reduced by developing a method for nearly optimal sampling of orientations. By using the quaternion representation of orientations, it is shown that the optimal sampling problem is equivalent to that of optimally distributing points on a fourdimensional sphere. In doing so, the number of orientation samples needed to achieve a desired indexing accuracy is significantly reduced. Orientation sets at a range of sizes are generated in this way for all Laue groups and are made available online for easy use.

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“…If the function is strictly completely monotonic, then the universally optimal solution is also unique. For points on the surface of a sphere, the only known universally optimal solutions are [39,40] N = 1-4, 6, and 12; that is, a single point, antipodal points, points forming an equilateral triangle on the equator, and tetrahedral, octahedral, and icosahedral arrangement of points. For our purposes, this means certain desired point arrangements (e.g.…”

Section: Discussionmentioning

confidence: 99%

Self-assembled clusters of spheres related to spherical codes

et al. 2012

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We consider the thermodynamically driven self-assembly of spheres onto the surface of a central sphere. This assembly process forms self-limiting, or terminal, anisotropic clusters (N -clusters) with well defined structures. We use Brownian dynamics to model the assembly of N -clusters varying in size from two to twelve outer spheres, and free energy calculations to predict the expected cluster sizes and shapes as a function of temperature and inner particle diameter. We show that the arrangements of outer spheres at finite temperatures are related to spherical codes, an ideal mathematical sequence of points corresponding to densest possible sphere packings. We demonstrate that temperature and the ratio of the diameters of the inner and outer spheres dictate cluster morphology and dynamics. We find that some N -clusters exhibit collective particle rearrangements, and these collective modes are unique to a given cluster size N . We present a surprising result for the equilibrium structure of a 5-cluster, which prefers an asymmetric square pyramid arrangement over a more symmetric arrangement. Our results suggest a promising way to assemble anisotropic building blocks from constituent colloidal spheres.

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Universally optimal distribution of points on spheres

Cited by 329 publications

References 46 publications

Counterintuitive ground states in soft-core models

Counterintuitive ground states in soft-core models

Improved orientation sampling for indexing diffraction patterns of polycrystalline materials

Self-assembled clusters of spheres related to spherical codes

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