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A339260
Decimal expansion of the maximum possible volume of a polyhedron with 8 vertices inscribed in the unit sphere.
4
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, 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, 8, 9, 9, 3, 7, 0, 0, 0, 2, 8, 9, 3, 8, 6, 5, 2, 7, 0, 3
OFFSET
1,2
COMMENTS
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.
LINKS
Joel D. Berman and Kitt Hanes, Volumes of Polyhedra Inscribed in the Unit Sphere in E3. Mathematische Annalen 188, 78-84 (1970).
Donald W. Grace, Search For Largest Polyhedra, Mathematics of Computation 17 (1963), pp. 197-199.
Matt Parker, The search for the biggest shape in the universe, YouTube video, 2024.
Hugo Pfoertner, Visualization of Polyhedron, (1999).
FORMULA
Equals sqrt((475 + 29*sqrt(145))/250).
EXAMPLE
1.8157161042244203975084949306331777890131009552754398376663729...
MATHEMATICA
RealDigits[Sqrt[(475 + 29*Sqrt[145])/250], 10, 120][[1]] (* Amiram Eldar, Jun 01 2023 *)
PROG
(PARI) sqrt((475+29*sqrt(145))/250)
CROSSREFS
Cf. A010527 (volume of double 5-pyramid), A081314, A081366, A122553 (volume of octahedron), A339259.
Sequence in context: A126585 A157289 A194097 * A226604 A154213 A353920
KEYWORD
nonn,cons
AUTHOR
Hugo Pfoertner, Nov 29 2020
STATUS
approved