%I
%S 24,12,48,20,8,12,16,4,120,4,24,12,24,4,6,2,8,2,4,6,4,2,2,20
%N Order of symmetry groups of n points on 3dimensional sphere with the volume enclosed by their convex hull maximized.
%C 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
%H Joel D. Berman and Kitt Hanes, <a href="http://gdz.sub.unigoettingen.de/dms/resolveppn/?PPN=GDZPPN002303426">Volumes of Polyhedra Inscribed in the Unit Sphere in E3</a>. Mathematische Annalen 188, 7884 (1970)
%H R. H. Hardin, N. J. A. Sloane and W. D. Smith, <a href="http://neilsloane.com/maxvolumes">Maximal Volume Spherical Codes</a>
%H Mutoh N., <a href="https://doi.org/10.1007/9783540444008_22">The Polyhedra of Maximal Volume Inscribed in the Unit Sphere and of Minimal Volume Circumscribed about the Unit Sphere</a>, in: Akiyama J., Kano M. (eds) Discrete and Computational Geometry. JCDCG 2002. Lecture Notes in Computer Science, vol 2866. Springer, Berlin, Heidelberg.
%H Hugo Pfoertner, <a href="http://www.randomwalk.de/sphere/volmax">Maximal Volume Arrangements of Points on Sphere.</a> Visualizations for n<=21.
%H Hugo Pfoertner, <a href="http://www.randomwalk.de/sphere/volmax/volmax.zip">Maximal Volume Arrangements: Archive</a>
%e 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 7pyramid proved optimal by Berman and Hanes has dihedral symmetry order 20.
%Y Number of distinct edges in convex hull: A081366. Symmetry groups for Tammes problem: A080865.
%K nonn,hard,more
%O 4,1
%A _Hugo Pfoertner_, Mar 19 2003
