A080865 -id:A080865 - cp's OEIS frontend

A081314 Order of symmetry groups of n points on 3-dimensional sphere with the volume enclosed by their convex hull maximized.

24, 12, 48, 20, 8, 12, 16, 4, 120, 4, 24, 12, 24, 4, 6, 2, 8, 2, 4, 6, 4, 2, 2, 20

Offset: 4

Hugo Pfoertner, Mar 19 2003

If more than one configuration with maximal volume exists for a given n, the one with the largest symmetry group is chosen. Berman and Hanes give optimality proofs for n<=8. Higher terms are only conjectures. An independent verification of the results by Hardin, Sloane and Smith has been performed by Pfoertner in 1992 for n<28. An archive of the results with improvements for n=23,24 added in 2003 is available at link. A conjectured continuation of the sequence starting with n=28 is: 12,6,2,6,120,2,4,4,2,20,4,12,24,12,20,4,8,2,2,2,4,1,24

			a(12)=120 because the order of the point group of the icosahedron, which is also the best known arrangement for the maximal volume problem is 120. a(7)=20 because the double 7-pyramid proved optimal by Berman and Hanes has dihedral symmetry order 20.

Joel D. Berman and Kitt Hanes, Volumes of Polyhedra Inscribed in the Unit Sphere in E3. Mathematische Annalen 188, 78-84 (1970)
R. H. Hardin, N. J. A. Sloane and W. D. Smith, Maximal Volume Spherical Codes
Mutoh N., The Polyhedra of Maximal Volume Inscribed in the Unit Sphere and of Minimal Volume Circumscribed about the Unit Sphere, in: Akiyama J., Kano M. (eds) Discrete and Computational Geometry. JCDCG 2002. Lecture Notes in Computer Science, vol 2866. Springer, Berlin, Heidelberg.
Hugo Pfoertner, Maximal Volume Arrangements of Points on Sphere. Visualizations for n<=21.
Hugo Pfoertner, Maximal Volume Arrangements: Archive

Number of distinct edges in convex hull: A081366. Symmetry groups for Tammes problem: A080865.

A084827 Maximum number of spheres of volume one that can be packed in a sphere of volume n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 3, 4, 4, 4, 6, 6, 6, 7, 8, 8, 9, 9, 10, 10, 12, 12, 13, 13, 13, 14, 14, 15, 15, 16, 16, 17, 18, 18, 19, 19, 19, 20, 21, 21, 21, 22, 22, 23, 23, 23, 25, 25, 26, 26, 26, 27, 28, 28, 29, 29, 30, 31, 31, 32, 33, 33, 34, 34, 35, 36, 36, 38, 38, 38, 38, 39, 39, 40, 40, 42, 42, 42, 43, 43, 44

Offset: 1

Hugo Pfoertner, Jun 09 2003

Higher terms of the sequence are only conjectures derived from numerical results. The first 12 arrangements are identical with the solutions of the Tammes problem (see A080865).

			a(10)=2 because a sphere of volume 10 is slightly too small to cover 3 mutually touching spheres of volume 1. a(27)=13 because the arrangement of 12 spheres plus one central sphere needs exactly a sphere with R=3*r to be contained.

Hugo Pfoertner, Table of n, a(n) for n = 1..132
Hugo Pfoertner, Densest packings of n equal spheres in a sphere of radius 1 (Table of largest possible radii)
Hugo Pfoertner, Numerical results for best packing of spheres in sphere.
Eric Weisstein's World of Mathematics, World of Mathematics: Sphere Packing.

Cf. A084828, A084829, A084824, A080865.

More terms from Hugo Pfoertner, May 09 2005

A242617 Decimal expansion of Kuijlaars-Saff constant, a constant related to Tammes' constants, Thomson's electron problem and Fekete points.

Original entry on oeis.org

5, 5, 3, 0, 5, 1, 2, 9, 3, 3, 5, 7, 5, 9, 5, 1, 8, 6, 7, 7, 9, 9, 5, 1, 0, 3, 7, 0, 8, 7, 1, 2, 4, 7, 7, 4, 5, 5, 0, 8, 0, 2, 8, 5, 7, 6, 0, 1, 9, 6, 6, 1, 7, 7, 6, 3, 3, 0, 4, 0, 7, 0, 9, 7, 0, 5, 9, 5, 3, 8, 7, 8, 8, 4, 0, 7, 7, 1, 2, 5, 4, 1, 6, 8, 7, 0, 5, 3, 7, 3, 2, 6, 3, 1, 6, 8, 2, 9, 1, 3

Offset: 0

Jean-François Alcover, May 19 2014

			-0.5530512933575951867799510370871247745508...

Steven R. Finch, Mathematical Constants, Cambridge, 2003, Section 8.8 p. 509.

A. B. J. Kuijlaars and E. B. Saff, Asymptotics for minimal discrete energy on the sphere
Eric Weisstein's World of Mathematics, Thomson Problem
Wikipedia, Thomson problem

Cf. A059750, A080865, A242088.

c = Sqrt[3]*Sqrt[Sqrt[3]/(8*Pi)]*Zeta[1/2]*(Zeta[1/2, 1/3] - Zeta[1/2, 2/3]); RealDigits[c, 10, 100] // First

sqrt(sqrt(27)/8/Pi)*zeta(1/2)*(zetahurwitz(1/2, 1/3) - zetahurwitz(1/2, 2/3)) \\ Charles R Greathouse IV, Jan 31 2018

sqrt(3)*sqrt(sqrt(3)/(8*Pi))*zeta(1/2)*(zeta(1/2, 1/3) - zeta(1/2, 2/3)).

A340918 Decimal expansion of largest angular separation (in radians) between 10 points on a unit sphere.

Original entry on oeis.org

1, 1, 5, 4, 4, 7, 9, 8, 3, 3, 4, 1, 9, 2, 7, 0, 7, 3, 7, 8, 3, 1, 9, 6, 1, 8, 4, 0, 4, 2, 3, 0, 2, 1, 1, 1, 4, 4, 8, 9, 3, 0, 0, 4, 8, 7, 3, 6, 3, 3, 4, 2, 5, 1, 2, 2, 4, 1, 4, 2, 1, 4, 4, 1, 7, 0, 4, 4, 4, 8, 3, 8, 1, 4, 0, 4, 9, 7, 4, 9, 6, 7, 1, 4, 5, 0, 0

Offset: 1

Hugo Pfoertner, Jan 30 2021

In his habilitation thesis from 1963, Ludwig Danzer provided an interval from 1.1544786 to 1.1544795 (rounded) for this value.

			1.1544798334192707378319618404230211144893004873633425122414214417...

Ludwig Danzer, Endliche Punktmengen auf der 2-Sphaere mit moeglichst grossem Minimalabstand. Habilitationsschrift, Universitaet Goettingen, 1963. See link for the English translation.

Paolo Xausa, Table of n, a(n) for n = 1..10000
Ludwig Danzer, Finite point-sets on S^2 with minimum distance as large as possible, Discrete Mathematics, Volume 60, June-July 1986, Pages 3-66, Table 1, page 63.
Hugo Pfoertner, Visualization of the optimal configuration, (2001).
Teruhisa Sugimoto and Masaharu Tanemura, Exact value of Tammes problem for N=10, arXiv:1509.01768 [math.MG], 12 Sep 2015.

Cf. A080865, A217695, A339262.

First[RealDigits[ArcTan[Sqrt[4/Sqrt[3]*Cos[ArcTan[Sqrt[3*229]/9]/3] + 3]], 10, 100]] (* Paolo Xausa, Feb 27 2025 *)

atan(sqrt((4/sqrt(3))*cos((1/3)*atan(sqrt(3*229)/9))+3))

atan(sqrt((4/sqrt(3))*cos((1/3)*atan(sqrt(3*229)/9)) + 3))

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.

A342559 Number of equal spheres setting a new density record in relation to the volume of the spherical layer that is occupied by the spheres when arranged touching the surface of a container sphere according to the criterion of maximizing their minimum mutual distance.

Original entry on oeis.org

2, 3, 4, 6, 8, 9, 10, 11, 12, 20, 24, 32, 38, 42, 44, 48

Offset: 1

Hugo Pfoertner, Apr 07 2021

This sequence is analogous to A084829, sharing the terms up to 12, but with the restriction that only arrangements are considered in which all small spheres touch the surface of the container sphere and the inner area remains empty. Formal proofs of optimality exist only for arrangements up to and including 14 and for 24 spheres, but the last improvements in the range of the specified terms were found before 1994. For references and links see A080865.
A conjectured continuation after the term 48 is 72, 78, 84, 92, 98, 120.
The linked illustration also shows a fitted curve estimating the minimum density achievable by optimal solutions of the Tammes problem for large n. The fitted equation is rho_min(n) = 0.565854 - 1/(0.566242*n + 2.67822). For comparison, consider the highest attainable density of spheres arranged in a flat hexagonal grid. This density is 0.604599788... = Pi * sqrt(3)/9. Achieving this density is made more difficult in the curved surface layer of a sphere because with large n there must always be 12 neighborhoods where the spheres packed in this layer can only have 5 nearest neighbors.

			  a(n)  Volume fraction in layer (rounded)
   2    0.25000
   3    0.30000
   4    0.36364
   6    0.42857
   8    0.43853
   9    0.45000
  10    0.45152
  11    0.46397
  12    0.50615
  20    0.51162
  24    0.52941
  32    0.53205
  38    0.53373
  42    0.53439
  44    0.54286
  48    0.54993

Hugo Pfoertner, Illustration of volume fraction of spheres in surface layer, (2021).

Cf. A080865, A084829, A121346.

A080866 Number of equal shortest edges in the solutions of the Tammes' problem.

Original entry on oeis.org

1, 3, 6, 6, 12, 12, 16, 18, 19, 25, 30, 24, 28

Offset: 2

Hugo Pfoertner, Feb 21 2003

A conjectured continuation of the sequence starting with n=15 would be: 30 32 34 34 34 39 40 42 43 60 48 46 52 52 54 63 60 66 66 68 66 72 66 72 76 78 81 85 82 88 84 91 89 120 96 102.

In the visualization given at the link the shortest edges are those drawn as golden color rods.

See under A080865.

Hugo Pfoertner, Arrangement of points on a sphere. Visualization of the best known solutions of the Tammes' problem.
Hugo Pfoertner, Table of edge lengths.

Cf. A080865.

cp's OEIS Frontend

A081314 Order of symmetry groups of n points on 3-dimensional sphere with the volume enclosed by their convex hull maximized.

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A084827 Maximum number of spheres of volume one that can be packed in a sphere of volume n.

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A242617 Decimal expansion of Kuijlaars-Saff constant, a constant related to Tammes' constants, Thomson's electron problem and Fekete points.

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A340918 Decimal expansion of largest angular separation (in radians) between 10 points on a unit sphere.

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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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A342559 Number of equal spheres setting a new density record in relation to the volume of the spherical layer that is occupied by the spheres when arranged touching the surface of a container sphere according to the criterion of maximizing their minimum mutual distance.

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A080866 Number of equal shortest edges in the solutions of the Tammes' problem.

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