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

Author

Hugo Pfoertner, Mar 19 2003

Keywords

• nonn
• hard
• more

Comments

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

Examples

			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.
		

Links

• 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

Crossrefs

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