A084827 -id:A084827 - cp's OEIS frontend

A080865 Order of symmetry groups of n points on 3-dimensional sphere with minimal distance between them maximized, also known as hostile neighbor or Tammes problem.

24, 12, 48, 6, 16, 12, 4, 10, 120, 8, 8

Offset: 4

Hugo Pfoertner, Feb 21 2003

If more than one best packing exists (this occurs for n = 15, 62, 76, 117, ...; see Buddenhagen, Kottwitz link) for a given n, the one with the largest symmetry group is chosen. A conjectured (except n=24) continuation of the sequence starting with n=15 would be: 3 16 4 2 2 12 1 1 1 24 3 2 4 1 1 6 5 6 3 2 1 4 2 24 1 3 1 10 1 2 1 2 1 24 2 12.

L. Fejes Toth, Lagerungen in der Ebene auf der Kugel und im Raum, 2nd. ed., Springer-Verlag, Berlin, Heidelberg 1972.

James Buddenhagen and D. A. Kottwitz, Multiplicity and Symmetry Breaking in (Conjectured) Densest Packings of Congruent Circles on a Sphere.
D. A. Kottwitz, The Densest Packing of Equal Circles on a Sphere, Acta Cryst. (1991). A47, 158-165
O. R. Musin and A. S. Tarasov, The strong thirteen spheres problem, Discrete Comput. Geom., 48 (2012), 128-141, arXiv:1002.1439 [math.MG], 2010-2012.
O. R. Musin and A. S. Tarasov, The Tammes problem for N=14, Experimental Mathematics, 24 (2015), 460-468, arXiv:1410.2536 [math.MG].
Hugo Pfoertner, Arrangement of points on a sphere. Visualization of the best known solutions of the Tammes problem.
K. Schuette and B. L. van der Waerden, Auf welcher Kugel haben 5, 6, 7, 8 oder 9 Punkte mit Mindestabstand Eins Platz?, Math. Annalen, 123 (1951), 96-124.
K. Schuette and B. L. van der Waerden, Auf welcher Kugel haben 5, 6, 7, 8 oder 9 Punkte mit Mindestabstand Eins Platz?, Math. Annalen, 123 (1951), 96-124.
N. J. A. Sloane, Library of 3-d packings

A080866 gives the number of shortest edges which make up the rigid framework of the arrangement.
Cf. A081314, A084827, A242088, A242617, A217695, A268487.
Cf. A342559 (point numbers where records of packing density occur).

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)).

Original entry on oeis.org

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

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

A080865 Order of symmetry groups of n points on 3-dimensional sphere with minimal distance between them maximized, also known as hostile neighbor or Tammes problem.

Views

Author

Keywords

Comments

References

Links

Crossrefs

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)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

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

Views

Author

Keywords

Comments

Links

Crossrefs

Extensions

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

Views

Author

Keywords

Comments

Links

Crossrefs

Extensions