cp's OEIS Frontend

A339263 Decimal expansion of the conjecturally maximum possible volume of a polyhedron with 11 vertices inscribed in the unit sphere.

%I A339263 #9 Aug 02 2025 11:35:57
%S A339263 2,3,5,4,6,3,4,4,9,5,0,6,8,6,1,5,2,0,3,2,3,6,8,8,0,5,9,2,6,3,8,9,2,6,
%T A339263 5,4,1,6,0,3,4,4,8,6,4,2,6,9,3,4,2,1,6,8,5,9,9,6,0,7,5,6,6,0,7,9,8,5,
%U A339263 4,5,8,3,1,4,8,1,5,5,5,3,1,5,0,1,9,4,5
%N A339263 Decimal expansion of the conjecturally maximum possible volume of a polyhedron with 11 vertices inscribed in the unit sphere.
%C A339263 The polyhedron (see linked illustration) with a symmetry group of order 4 has a vertex in the north pole on its axis of symmetry. The remaining 10 vertices are diametrically opposite in pairs relative to this axis of symmetry. The polar vertex has vertex degree 6. 8 vertices have vertex degree 5. 2 vertices have vertex degree 4.
%C A339263 This allocation seems to be the best possible approximation of a medial distribution of the vertex degrees, which is a known necessary condition for maximum volume. Of the 25 possible triangulations with vertex degree >= 4, all the others have more than 2 vertices with vertex degree 4, which leads to more pointed corners and therefore smaller volumes.
%H A339263 R. H. Hardin, N. J. A. Sloane and W. D. Smith, <a href="http://neilsloane.com/maxvolumes">Maximal Volume Spherical Codes</a>.
%H A339263 Hugo Pfoertner, <a href="http://www.randomwalk.de/sphere/volmax/pages/11.htm">Visualization of Polyhedron</a>, (1999).
%H A339263 Hugo Pfoertner, <a href="http://www.randomwalk.de/sphere/volmax/videos/11c.mp4">Number of edges incident with the 11 vertices</a>, video (2021).
%e A339263 2.35463449506861520323688059263892654160344864269342168599607566...
%Y A339263 Cf. A010527 (volume of double 5-pyramid), A081314, A081366, A122553 (volume of octahedron), A339259, A339260, A339261, A339262.
%K A339263 nonn,cons
%O A339263 1,1
%A A339263 _Hugo Pfoertner_, Dec 07 2020