A084825 -id:A084825 - cp's OEIS frontend

A084824 Maximum number of spheres of diameter one that can be packed in a cube of volume n (i.e., with edge length n^(1/3)).

1, 1, 1, 2, 4, 4, 5, 8, 8, 8, 9, 9, 10, 11, 14, 14, 14, 15, 18, 18, 19, 19, 21, 21, 23, 24, 27, 27, 27, 27, 32, 32, 32, 33

Offset: 1

Hugo Pfoertner, Jun 12 2003

Higher sequence terms are only conjectures found by numerical experimentation.

			a(5) = 4 because a cube of edge length 5^(1/3) = 1.7099759 is large enough to contain 4 spheres arranged as a tetrahedron, which requires a minimum enclosing cube of edge length 1+sqrt(2)/2 = 1.70710678.

Dave Boll, Optimal Packing of Circles and Spheres
Thierry Gensane, Dense Packings of Equal Spheres in a Cube, The Electronic Journal of Combinatorics 11 (2004), #R33.
M. Goldberg, On the Densest Packing of Equal Spheres in a Cube, Math. Mag. 44, 199-208, 1971.
Hugo Pfoertner, Best packing of equal spheres in a cube. Numerical results.
Hugo Pfoertner, Densest Packings of Equal Spheres in a Cube. Visualizations.
J. Schaer, On the Densest Packing of Spheres in a Cube, Can. Math. Bul. 9, 265-270, 1966.

Cf. A084825, A084826, A084827, A084616.

Corrected erroneous a(14) and extended to a(34) by Hugo Pfoertner, including results from Thierry Gensane, Jun 23 2011

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

cp's OEIS Frontend

A084824 Maximum number of spheres of diameter one that can be packed in a cube of volume n (i.e., with edge length n^(1/3)).

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