Research problem

Tóth, László

doi:10.1007/bf01876492

Cited by 4 publications

(3 citation statements)

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“…But until someone extends Hales's result to higher dimensions, we have no proof that any packing in a dimension greater than three is optimal. these are expressed differently in animals that have distinct patterns of segmental specialization [4][5][6] . Similarities in the expression of these genes have been used as evidence for a common origin (homology) of the corre-sponding specialized types of segment 4 .…”

Section: Figure 1 Cannonballs Stacked In a Face-centredcubic Lattice mentioning

confidence: 99%

Kepler's conjecture confirmed

Sloane

1998

Nature

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Section: Figure 1 Cannonballs Stacked In a Face-centredcubic Lattice mentioning

confidence: 99%

Kepler's conjecture confirmed

Sloane

1998

Nature

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“…Surprisingly, the densest packing of a number of spheres N within their convex hull is not always a compact cluster of spheres. On the contrary, in 1975 the mathematician Fejes Tóth conjectured 14 that for dimensions d ≥ 5 the densest packing is the one where the centers of the spheres are aligned along a straight line, resulting in a so-called sausage configuration. This conjecture, supported by other studies 15,16 , was initially proven true only for d ≥ 13387 17 and subsequently for d ≥ 42 18 .…”

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confidence: 99%

A colloidal viewpoint on the sausage catastrophe and the finite sphere packing problem

Marín-Aguilar,

Camerin,

van der Ham

et al. 2023

Nat Commun

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It is commonly believed that the most efficient way to pack a finite number of equal-sized spheres is by arranging them tightly in a cluster. However, mathematicians have conjectured that a linear arrangement may actually result in the densest packing. Here, our combined experimental and simulation study provides a physical realization of the finite sphere packing problem by studying arrangements of colloids in a flaccid lipid vesicle. We map out a state diagram displaying linear, planar, and cluster conformations of spheres, as well as bistable states which alternate between cluster-plate and plate-linear conformations due to membrane fluctuations. Finally, by systematically analyzing truncated polyhedral packings, we identify clusters of 56 ≤ N ≤ 70 number of spheres, excluding N = 57 and 63, that pack more efficiently than linear arrangements.

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“…

THE SAUSAGE CONJECTURE (L. Fejes T6th, [8]). Let P be a packing of n __> 2 unit balls in E d, d __> 5.

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confidence: 99%

The sausage conjecture holds for convex hulls of moderately bent sausages

Dekster

1996

Acta Math Hung

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THE SAUSAGE CONJECTURE (L. Fejes T6th, [8]). Let P be a packing of n __> 2 unit balls in E d, d __> 5. Denote by Vp the volume of the convex hull of these n balls and let nk be the volume of a k-dimensional unit ball. Then(1)Vp >= 2~d-l(n-1)+ rid. The equality holds here only if the centers of the balls of P are equally spaced on a line at distance 2. (Then their convex hull resembles a sausage.) This conjecture has been proved in a few particular cases. Betke, Henk and Wills [4] proved it for dimensions d __> 13387. By improving the method, Henk [10] recently improved the bound to d __> 45. In a still unpublished paper, Betke and Henk improved the lower bound on d to 42. Let Cn be the convex hull of the centers of the balls of P. The conjecture holds also in the following cases: dimC,~ _<_ ~(d-1), [3]; dim C~ <= 9 and dim C,~ __< d -1, [1]; Cn is a lattice zonotope, [2];Cn is a regular simplex and the balls are centered at its vertices, [7]. Another particular case is mentioned in [9, (2.7)]. Some related results are described there, too.The Theorem here proves the conjecture under yet other restrictions on the location of the balls of P. To specify the results, we need a few definitions. Let G = (V, E) be a polygonal network in E d with a set V of n => 2 distinct vertices (points) and a set E of edges, i.e. segments connecting some pairs of the vertices. The edges can cross or overlap. If a, b, c, d E V, ab E E and c, d E ab, then the segments cd, ca, cb are not necessarily in E. Thus G is a realization of a graph in E d (not necessarily an embedding). We will use for G the standard graph terminology.Suppose that (i) If segments ab, bc E E, then the angle Aabc >_ ~r/2. (ii) If a, b are adjacent vertices of G, then the distance ab (in E d) is at least 2. If they are not adjacent, then ab >__ 2V"2.

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Research problem

Cited by 4 publications

References 6 publications

Kepler's conjecture confirmed

Kepler's conjecture confirmed

A colloidal viewpoint on the sausage catastrophe and the finite sphere packing problem

The sausage conjecture holds for convex hulls of moderately bent sausages

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