A121346 -id:A121346 - cp's OEIS frontend

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

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

A084829 Best packing of m>1 equal spheres in a sphere setting a new density record.

Original entry on oeis.org

2, 3, 4, 6, 8, 9, 11, 12, 18, 21, 25, 30, 31, 32, 33, 34, 35, 36, 38, 49, 51, 53, 56, 59, 60, 61

Offset: 1

Hugo Pfoertner, Jun 12 2003

All terms beyond m=9 are only conjectures found by numerical experimentation. The density is defined as the fraction of the volume of the large sphere occupied by the small spheres. For 2 spheres the density is 0.25. The first known configuration with density exceeding 0.5 occurs for 31 spheres.

See the E. Specht link for latest results. - Eduard Baumann, Jan 03 2024

Dave Boll, Optimal Packing of Circles and Spheres.
WenQi Huang and Liang Yu, Serial Symmetrical Relocation Algorithm for the Equal Sphere Packing Problem, arXiv:1202.4149 [cs.DM], 2012.
Eckard Specht, The best known packings of equal spheres in a sphere, July 2023.
Hugo Pfoertner, Numerical results for best packing of spheres in sphere.
Hugo Pfoertner, Densest Packings of n Equal Spheres in a Sphere of Radius 1 Largest Possible Radii.
Eric Weisstein's World of Mathematics, Sphere Packing.
Jianrong Zhou, Shuo Ren, Kun He, Yanli Liu, and Chu-Min Li, An Efficient Solution Space Exploring and Descent Method for Packing Equal Spheres in a Sphere, arXiv:2305.10023 [cs.CG], 2023.

Cf. A084827, A084826, A084644, A084828, A121346.

Inserted missing term 30, added comment with conjectured next terms and updated links by Hugo Pfoertner, Jun 24 2011

More terms from Hugo Pfoertner, Aug 25 2013

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

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A084829 Best packing of m>1 equal spheres in a sphere 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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