cp's OEIS Frontend

A339260 Decimal expansion of the maximum possible volume of a polyhedron with 8 vertices inscribed in the unit sphere.

%I A339260 #24 Aug 02 2025 11:11:27
%S A339260 1,8,1,5,7,1,6,1,0,4,2,2,4,4,2,0,3,9,7,5,0,8,4,9,4,9,3,0,6,3,3,1,7,7,
%T A339260 7,8,9,0,1,3,1,0,0,9,5,5,2,7,5,4,3,9,8,3,7,6,6,6,3,7,2,9,1,6,9,1,8,4,
%U A339260 8,9,9,3,7,0,0,0,2,8,9,3,8,6,5,2,7,0,3
%N A339260 Decimal expansion of the maximum possible volume of a polyhedron with 8 vertices inscribed in the unit sphere.
%C A339260 Berman and Hanes (see link, page 81) proved in 1970 that an arrangement of 8 points on the surface of a sphere with 4 points with node degree 4 and 4 points with node degree 5 is the one with a maximum volume of their convex hull.
%H A339260 Joel D. Berman and Kitt Hanes, <a href="http://gdz.sub.uni-goettingen.de/dms/resolveppn/?PPN=GDZPPN002303426">Volumes of Polyhedra Inscribed in the Unit Sphere in E3</a>. Mathematische Annalen 188, 78-84 (1970).
%H A339260 Donald W. Grace, <a href="https://doi.org/10.1090/S0025-5718-63-99183-X">Search For Largest Polyhedra</a>, Mathematics of Computation 17 (1963), pp. 197-199.
%H A339260 Matt Parker, <a href="https://www.youtube.com/watch?v=XZy3rXr2yeM">The search for the biggest shape in the universe</a>, YouTube video, 2024.
%H A339260 Hugo Pfoertner, <a href="http://www.randomwalk.de/sphere/volmax/pages/08.htm">Visualization of Polyhedron</a>, (1999).
%H A339260 Hugo Pfoertner, <a href="http://www.randomwalk.de/sphere/volmax/videos/08c.mp4">Number of edges incident with the 8 vertices</a>, video (2021).
%F A339260 Equals sqrt((475 + 29*sqrt(145))/250).
%e A339260 1.8157161042244203975084949306331777890131009552754398376663729...
%t A339260 RealDigits[Sqrt[(475 + 29*Sqrt[145])/250], 10, 120][[1]] (* _Amiram Eldar_, Jun 01 2023 *)
%o A339260 (PARI) sqrt((475+29*sqrt(145))/250)
%Y A339260 Cf. A010527 (volume of double 5-pyramid), A081314, A081366, A122553 (volume of octahedron), A339259.
%K A339260 nonn,cons
%O A339260 1,2
%A A339260 _Hugo Pfoertner_, Nov 29 2020