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

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

Author

Hugo Pfoertner, Nov 29 2020

Keywords

• nonn
• cons

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.

Examples

			1.8157161042244203975084949306331777890131009552754398376663729...
		

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).
• Hugo Pfoertner, Number of edges incident with the 8 vertices, video (2021).

Crossrefs

Cf. A010527 (volume of double 5-pyramid), A081314, A081366, A122553 (volume of octahedron), A339259.

Programs

Mathematica

RealDigits[Sqrt[(475 + 29*Sqrt[145])/250], 10, 120][[1]] (* Amiram Eldar, Jun 01 2023 *)

PARI

sqrt((475+29*sqrt(145))/250)

Formula

Equals sqrt((475 + 29*sqrt(145))/250).