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

1, 2, 13, 32, 68

Offset

1,2

Comments

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

Links

Table of n, a(n) for n=1..5.

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.

Crossrefs

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

Cf. A023393 (2D).

Sequence in context: A285096 A177455 A185950 * A100512 A051474 A062708

Adjacent sequences: A084825 A084826 A084827 * A084829 A084830 A084831

Keyword

hard,more,nonn

Author

Hugo Pfoertner, Jun 12 2003

Extensions

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

Status

approved