A084826 -id:A084826 - 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

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

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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