A084829 -id:A084829 - cp's OEIS frontend

A084827 Maximum number of spheres of volume one that can be packed in a sphere of volume n.

1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 3, 4, 4, 4, 6, 6, 6, 7, 8, 8, 9, 9, 10, 10, 12, 12, 13, 13, 13, 14, 14, 15, 15, 16, 16, 17, 18, 18, 19, 19, 19, 20, 21, 21, 21, 22, 22, 23, 23, 23, 25, 25, 26, 26, 26, 27, 28, 28, 29, 29, 30, 31, 31, 32, 33, 33, 34, 34, 35, 36, 36, 38, 38, 38, 38, 39, 39, 40, 40, 42, 42, 42, 43, 43, 44

Offset: 1

Hugo Pfoertner, Jun 09 2003

Higher terms of the sequence are only conjectures derived from numerical results. The first 12 arrangements are identical with the solutions of the Tammes problem (see A080865).

			a(10)=2 because a sphere of volume 10 is slightly too small to cover 3 mutually touching spheres of volume 1. a(27)=13 because the arrangement of 12 spheres plus one central sphere needs exactly a sphere with R=3*r to be contained.

Hugo Pfoertner, Table of n, a(n) for n = 1..132
Hugo Pfoertner, Densest packings of n equal spheres in a sphere of radius 1 (Table of largest possible radii)
Hugo Pfoertner, Numerical results for best packing of spheres in sphere.
Eric Weisstein's World of Mathematics, World of Mathematics: Sphere Packing.

Cf. A084828, A084829, A084824, A080865.

More terms from Hugo Pfoertner, May 09 2005

A084828 Maximum number of spheres of radius one that can be packed in a sphere of radius n.

Original entry on oeis.org

1, 2, 13, 32, 68

Offset: 1

Hugo Pfoertner, Jun 12 2003

a(4) and a(5) are experimental values. Although A121346(5) claims a lower bound of a(5)=68, it is conjectured from extensive numerical search that this value is unachievable and therefore a(5)=67.
The conjecture a(5)=67 was proved wrong by Yu Liang, who found an arrangement of 68 spheres of radius 1 fitting into a sphere of radius 5.
Lower bounds for the next terms are a(6)>=122 and a(7)>=198. See E. Specht's webpage for latest data. - Hugo Pfoertner, Jan 22 2024

Sen Bai, X. Bai, X. Che, and X. Wei, Maximal Independent Sets in Heterogeneous Wireless Ad Hoc Networks, IEEE Transactions on Mobile Computing (Volume: 15, Issue: 8, Aug 01 2016), pp. 2023-2033.
Dave Boll, Optimal Packing of Circles and Spheres.
Sunil K. Chebolu, Packing Moons Inside the Earth, arXiv:2006.00603 [physics.pop-ph], 2020.
WenQi Huang and Liang Yu, A Quasi Physical Method for the Equal Sphere Packing Problem, in 2011 IEEE 10th International Conference on Trust, Security and Privacy in Computing and Communications.
WenQi Huang and Liang Yu, Serial Symmetrical Relocation Algorithm for the Equal Sphere Packing Problem, arXiv preprint arXiv:1202.4149 [cs.DM], 2012. - From _N. J. A. Sloane_, Jun 14 2012
Hugo Pfoertner, Numerical results for best packing of spheres in sphere.
Hugo Pfoertner, Densest Packing of Spheres in a Sphere. Java visualization.
Eckhard Specht, The best known packings of equal spheres in a sphere.
Yu Liang, Coordinates of sphere centers of 68 spheres of radius 0.20000222, fitting into a container of radius 1. Private communication, Aug 22 2011.

Cf. A121346 (conjectured lower bounds), A084827, A084829, A084825.

Cf. A023393 (2D).

Comment and links edited, a(5) from Hugo Pfoertner, Jun 23 2011

a(5) corrected, based on private communication from Yu Liang, by Hugo Pfoertner, Aug 24 2011

A084826 Best packing of m>1 equal spheres in a cube setting a new density record.

Original entry on oeis.org

2, 3, 4, 6, 7, 8, 31, 32, 48, 60, 61, 62, 63

Offset: 1

Hugo Pfoertner, Jun 12 2003

The terms >=31 are only conjectures found by numerical experimentation. In the table given at the Pfoertner link, the densities are given relative to the density of the cubic lattice packing (Pi/6). The first known arrangement with higher density than that of the cubic lattice packing was found for m=31 spheres. In the region 8

See under A084824.

Dave Boll, Packing spheres in cubes
Hugo Pfoertner, Best packing of equal spheres in a cube. Numerical results.
Hugo Pfoertner, Densest packing of 32 spheres in a cube. Video.
Hugo Pfoertner, Densest packing of 48 spheres in a cube. Video.
Hugo Pfoertner, Densest packing of 63 spheres in a cube. Video.

Cf. A084824, A084829, A051657.

More terms from Hugo Pfoertner, Oct 03 2015

A342559 Number of equal spheres setting a new density record in relation to the volume of the spherical layer that is occupied by the spheres when arranged touching the surface of a container sphere according to the criterion of maximizing their minimum mutual distance.

Original entry on oeis.org

2, 3, 4, 6, 8, 9, 10, 11, 12, 20, 24, 32, 38, 42, 44, 48

Offset: 1

Hugo Pfoertner, Apr 07 2021

This sequence is analogous to A084829, sharing the terms up to 12, but with the restriction that only arrangements are considered in which all small spheres touch the surface of the container sphere and the inner area remains empty. Formal proofs of optimality exist only for arrangements up to and including 14 and for 24 spheres, but the last improvements in the range of the specified terms were found before 1994. For references and links see A080865.
A conjectured continuation after the term 48 is 72, 78, 84, 92, 98, 120.
The linked illustration also shows a fitted curve estimating the minimum density achievable by optimal solutions of the Tammes problem for large n. The fitted equation is rho_min(n) = 0.565854 - 1/(0.566242*n + 2.67822). For comparison, consider the highest attainable density of spheres arranged in a flat hexagonal grid. This density is 0.604599788... = Pi * sqrt(3)/9. Achieving this density is made more difficult in the curved surface layer of a sphere because with large n there must always be 12 neighborhoods where the spheres packed in this layer can only have 5 nearest neighbors.

			  a(n)  Volume fraction in layer (rounded)
   2    0.25000
   3    0.30000
   4    0.36364
   6    0.42857
   8    0.43853
   9    0.45000
  10    0.45152
  11    0.46397
  12    0.50615
  20    0.51162
  24    0.52941
  32    0.53205
  38    0.53373
  42    0.53439
  44    0.54286
  48    0.54993

Hugo Pfoertner, Illustration of volume fraction of spheres in surface layer, (2021).

Cf. A080865, A084829, A121346.

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A084827 Maximum number of spheres of volume one that can be packed in a sphere of volume n.

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A084828 Maximum number of spheres of radius one that can be packed in a sphere of radius n.

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A084826 Best packing of m>1 equal spheres in a cube setting a new density record.

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A342559 Number of equal spheres setting a new density record in relation to the volume of the spherical layer that is occupied by the spheres when arranged touching the surface of a container sphere according to the criterion of maximizing their minimum mutual distance.

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