A242617 -id:A242617 - 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).

A242088 Number of edges in the convex hull of a rigorous solution to Thomson's problem for n points.

Original entry on oeis.org

0, 0, 1, 3, 6, 9, 12

Offset: 0

Jonathan Sondow, May 04 2014

Thomson’s problem is to determine the stable equilibrium configuration(s) of n particles confined to the surface of a sphere and repelling each other by an inverse square force.
Rigorous solutions are known only for n <= 6 and n = 12, with a(12) = 30.
Non-rigorous solutions are given in Wikipedia for all n <= 460. The least non-monotonic pair is 63 > 60 for n = 23 and 24, respectively.

			For n = 0 or 1 points, the convex hull is empty or a point, so there are no edges and a(0) = a(1) = 0.

Kevin Brown, Min-Energy Configurations of Electrons On A Sphere
Eric Weisstein's MathWorld, Thomson Problem
Wikipedia, Thomson problem

Cf. A008486, A008585, A080865, A242617, A268487.

a(n) <= n(n-1)/2 = (n choose 2).

a(n) <= 3*n-6 = A008585(n-2) for n >= 3, since a solution to Thomson's problem gives a planar graph, which has 3*n-6 edges if it is maximal (see A008486 comments). - Jonathan Sondow, Mar 03 2018 answering a question by Joseph Wheat.

A033177 Number of distinct distances between n electrons in minimal energy configuration on a sphere.

Original entry on oeis.org

0, 1, 1, 1, 3, 2, 4, 4, 6, 7, 20, 3, 28, 9, 21, 12, 14, 18, 52, 26, 69, 17, 48, 16, 160, 168

Offset: 1

K. S. Brown (ksbrown(AT)seanet.com)

Most of the higher terms of this sequence just refer to the best configurations known today.

How is this defined if the optimal configuration is not unique? - N. J. A. Sloane, Feb 27 2009

			See Brown's web page in LINKS.

K. S. Brown, Min-Energy Configurations of Electrons On A Sphere, MathPages.
R. H. Hardin, N. J. A. Sloane & W. D. Smith, Table of best configurations presently known, Feb 1994.
Wikipedia, Thomson Problem.

Cf. A133491, A242617.

A133491 Order of the symmetry group of the (in some cases conjectural) minimal-energy configuration of n identical charged particles confined to the surface of a sphere.

Original entry on oeis.org

12, 24, 12, 48, 20, 16, 12, 16, 4, 120, 4

Offset: 3

Keenan Pepper, Nov 30 2007

a(0), a(1) and a(2) are all infinite, because their symmetry groups are continuous (they contain rotations with arbitrary angles). Actual symmetry groups: 3 D_{3h}, 4 T_{d}, 5 D_{3h}, 6 O_{d}, 7 D_{5h}, 8 D_{4d}, 9 D_{3h}, 10 D_{4h}, 11 D_{1h}, 12 I_{d}, 13 D_{1h}.

			a(3)=12 because the minimal-energy configuration of 3 charged particles on a sphere is an equilateral triangle on the equator, which has symmetry group D_3h of order 12.

K. S. Brown, Min-Energy Configurations of Electrons On A Sphere, MathPages.
R. H. Hardin, N. J. A. Sloane and W. D. Smith, Minimal Energy Configurations of Points on a Sphere
Wikipedia, Thomson Problem.

Cf. A033177, A242617.

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.

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A242088 Number of edges in the convex hull of a rigorous solution to Thomson's problem for n points.

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A033177 Number of distinct distances between n electrons in minimal energy configuration on a sphere.

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A133491 Order of the symmetry group of the (in some cases conjectural) minimal-energy configuration of n identical charged particles confined to the surface of a sphere.

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