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A081314 Order of symmetry groups of n points on 3-dimensional sphere with the volume enclosed by their convex hull maximized. 3
24, 12, 48, 20, 8, 12, 16, 4, 120, 4, 24, 12, 24, 4, 6, 2, 8, 2, 4, 6, 4, 2, 2, 20 (list; graph; refs; listen; history; text; internal format)



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


Table of n, a(n) for n=4..27.

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

Hugo Pfoertner, Maximal Volume Arrangements of Points on Sphere. Visualizations for n<=21.

Hugo Pfoertner, Maximal Volume Arrangements: Archive


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.


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

Sequence in context: A307267 A080865 A040555 * A119872 A002550 A075605

Adjacent sequences:  A081311 A081312 A081313 * A081315 A081316 A081317




Hugo Pfoertner, Mar 19 2003



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