Das Problem der dreizehn Kugeln

Schütte, Kurt

doi:10.1007/bf01343127

Cited by 133 publications

(102 citation statements)

References 2 publications

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“…A pivotal role in the analytic modeling of granular matter is played by the densest local structure [2,3]. This structure coincides with the solution of the kissing problem in mathematics [4,5], so we need to maximize the number of spherical particles simultaneously in contact with a central one. The icosahedral cluster, depicted in Fig.…”

Section: Introductionmentioning

confidence: 92%

Densest Local Structures of Uniaxial Ellipsoids

2016

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Connecting the collective behavior of disordered systems with local structure on the particle scale is an important challenge, for example, in granular and glassy systems. Compounding complexity, in many scientific and industrial applications, particles are polydisperse, aspherical, or even of varying shape. Here, we investigate a generalization of the classical kissing problem in order to understand the local building blocks of packings of aspherical grains. We numerically determine the densest local structures of uniaxial ellipsoids by minimizing the Set Voronoi cell volume around a given particle. Depending on the particle aspect ratio, different local structures are observed and classified by symmetry and Voronoi coordination number. In extended disordered packings of frictionless particles, knowledge of the densest structures allows us to rescale the Voronoi volume distributions onto the single-parameter family of k-Gamma distributions. Moreover, we find that approximate icosahedral clusters are found in random packings, while the optimal local structures for more aspherical particles are not formed.

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Section: Introductionmentioning

confidence: 92%

Densest Local Structures of Uniaxial Ellipsoids

2016

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“…Let r(B) be the radius of a ball B ∈ P . By a famous result ( [6], [4]), it is impossible for more than 12 unit balls with disjoint interiors to kiss a unit ball B. If C kisses B and r(C) > 1 = r(B), then C contains a (unique) unit ball that kisses B.…”

Section: The Upper Bound Theorem If P Is a Finite Ball Packing Inmentioning

confidence: 99%

Average kissing numbers for non-congruent sphere packings

Kuperberg

Schramm

1994

Mathematical Research Letters

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“…When D = 2, the maximum feasible N is of course 6 (hexagonal lattice). When D = 3, the maximum feasible N was conjectured by Newton to be 12 and by Gregory to be 13 (Newton was proven right 180 years later [58] …”

Section: For Any H ≤ G If G(x)mentioning

confidence: 99%

Symmetry in Mathematical Programming

Liberti

2011

Mixed Integer Nonlinear Programming

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Abstract. Symmetry is mainly exploited in mathematical programming in order to reduce the computation times of enumerative algorithms. The most widespread approach rests on: (a) finding symmetries in the problem instance; (b) reformulating the problem so that it does not allow some of the symmetric optima; (c) solving the modified problem. Sometimes (b) and (c) are performed concurrently: the solution algorithm generates a sequence of subproblems, some of which are recognized to be symmetrically equivalent and either discarded or treated differently. We review symmetry-based analyses and methods for Linear Programming, Integer Linear Programming, Mixed-Integer Linear Programming and Semidefinite Programming. We then discuss a method (introduced in [35]) for automatically detecting symmetries of general (nonconvex) Nonlinear and Mixed-Integer Nonlinear Programming problems and a reformulation based on adjoining symmetry breaking constraints to the original formulation. We finally present a new theoretical and computational study of the formulation symmetries of the Kissing Number Problem.

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Das Problem der dreizehn Kugeln

Cited by 133 publications

References 2 publications

Densest Local Structures of Uniaxial Ellipsoids

Densest Local Structures of Uniaxial Ellipsoids

Average kissing numbers for non-congruent sphere packings

Symmetry in Mathematical Programming

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