Packing of spheres in spaces of constant curvature

Böröczky, Károly J.

doi:10.1007/bf01902361

Cited by 165 publications

(310 citation statements)

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“…Coxeter's bounds are based on the conjecture that equal size spherical caps on a sphere can be packed no denser than packing where the Delaunay triangulation with vertices at the centers of caps consists of regular simplices. This conjecture was proved by Böröczky in 1978 [5].…”

Section: Introductionmentioning

confidence: 78%

The kissing number in four dimensions

Musin¹

2008

Ann. Math.

122

129

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The kissing number problem asks for the maximal number k(n) of equal size nonoverlapping spheres in n-dimensional space that can touch another sphere of the same size. This problem in dimension three was the subject of a famous discussion between Isaac Newton and David Gregory in 1694. In three dimensions the problem was finally solved only in 1953 by Schütte and van der Waerden.In this paper we present a solution of a long-standing problem about the kissing number in four dimensions. Namely, the equality k(4) = 24 is proved. The proof is based on a modification of Delsarte's method.

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Section: Introductionmentioning

confidence: 78%

The kissing number in four dimensions

Musin¹

2008

Ann. Math.

122

129

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“…We obtain a better estimate on volume by combining this cusp estimate with Böröczky's lower bound for the density of a horoball packing in hyperbolic space [3]. By his result, the volume of a maximal cusp neighborhood in a hyperbolic 3-manifold M is at most 3 Deformation through cone manifolds…”

Section: Lemma 24mentioning

confidence: 94%

Volumes of highly twisted knots and links

Purcell

2007

Algebr. Geom. Topol.

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We show that for a large class of knots and links with complements in S 3 admitting a hyperbolic structure, we can determine bounds on the volume of the link complement from combinatorial information given by a link diagram. Specifically, there is a universal constant C such that if a knot or link admits a prime, twist reduced diagram with at least 2 twist regions and at least C crossings per twist region, then the link complement is hyperbolic with volume bounded below by 3.3515 times the number of twist regions in the diagram. C is at most 113. 57M25, 57M501 IntroductionGiven a diagram of a knot or link, our goal is to determine geometric information about the complement of that link in S 3 . In particular, if the complement admits a hyperbolic structure, then by Mostow-Prasad rigidity that structure is unique. We ought to be able to make explicit statements about the geometry of this link complement, including statements about its volume. However, such results based purely on a diagram seem to be rare. For particular classes of knots and links, other results on volume have been determined. Lackenby proved that in the special case in which a knot or link is alternating, the volume of the complement is bounded above and below by the twist number of a diagram [10]. In fact, the upper bound is valid for all knots, not just alternating. This upper bound was further improved by Agol and D Thurston in an appendix to Lackenby's paper. Additionally, they found a sequence of links with volume approaching the upper bound.Recently, the lower bound has been improved by work of Agol, Storm and Thurston [2]. The proof of this result still requires that the links in question be alternating, however.

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“…Theorem 1 in [Bo2] proves that the Voronoi cells in x have the highest relative density among Voronoi cells of all packings in S R . From this and Proposition 3 it follows that µ x (and x) is optimally dense.…”

Section: Proof Of Propositionmentioning

confidence: 99%

“…While the study of densest packings of spheres in Euclidean space has made impressive gains in recent years [H], the analogous study in hyperbolic space has been held back at a fundamental level; there has not been a convincing approach to define what one should mean by "densest packing of spheres" in hyperbolic space [Bo2], [Fej], [FeK], [FKK]; see especially the discussion on pp. 831-834 of [FeK].…”

Section: Introductionmentioning

confidence: 99%