A339262 Decimal expansion of the conjecturally maximum possible volume of a polyhedron with 10 vertices inscribed in the unit sphere.

2, 2, 1, 8, 7, 1, 1, 1, 3, 1, 5, 4, 5, 3, 9, 9, 4, 0, 3, 2, 4, 7, 2, 8, 2, 7, 5, 1, 1, 2, 8, 4, 1, 7, 0, 1, 3, 8, 1, 0, 7, 2, 5, 3, 7, 4, 6, 6, 3, 3, 4, 4, 3, 8, 1, 7, 5, 0, 0, 4, 9, 0, 8, 4, 2, 0, 1, 0, 0, 8, 1, 2, 7, 9, 9, 0, 9, 1, 8, 1, 4, 8, 8, 4, 6, 3, 3

Offset

1,1

Comments

The polyhedron (see linked illustration) has vertices at the poles and two square rings of vertices rotated by Pi/4 against each other, with a polar angle of approx. +-62.89908285 degrees against the poles. The polyhedron is completely described by this angle and its order 16 symmetry. It would be desirable to know a closed formula representation of this angle and the volume.

The polar angle is +-acos((1 - sqrt(2) + sqrt(6-2*sqrt(2)))/3). - Natalia L. Skirrow, Apr 11 2026

Links

Paolo Xausa, Table of n, a(n) for n = 1..10000

R. H. Hardin, N. J. A. Sloane, and W. D. Smith, Maximal Volume Spherical Codes.

Hugo Pfoertner, Visualization of Polyhedron, (1999).

Hugo Pfoertner, Number of edges incident with the 10 vertices, video (2021).

Natalia L. Skirrow, note on A339262.

Formula

Equals (2/3)^4 * (3 + sqrt(2) + (3-sqrt(2))*sqrt(10 + 6*sqrt(2))), and satisfies minimal polynomial 65536 + 147456*x - 59904*x^2 - 139968*x^3 + 59049*x^4. - Natalia L. Skirrow, Apr 11 2026

Example

2.218711131545399403247282751128417013810725374663344381750049084201...

Mathematica

First[RealDigits[16*(Sqrt[2] + Sqrt[6*Sqrt[2] + 38] + 3)/81, 10, 100]] (* Paolo Xausa, Apr 17 2026 *)

Crossrefs

Cf. A010527 (volume of double 5-pyramid), A081314, A081366, A122553 (volume of octahedron), A339259, A339260, A339261, A339263.

Sequence in context: A096440 A181738 A121350 * A198569 A135080 A394330

Adjacent sequences: A339259 A339260 A339261 * A339263 A339264 A339265

Keyword

nonn,cons

Author

Hugo Pfoertner, Dec 07 2020

Status

approved