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

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, 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, 4, 5, 8, 3, 1, 4, 8, 1, 5, 5, 5, 3, 1, 5, 0, 1, 9, 4, 5

Offset: 1

Author

Hugo Pfoertner, Dec 07 2020

Keywords

• nonn
• cons

Comments

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.

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.

Examples

			2.35463449506861520323688059263892654160344864269342168599607566...
		

Links

• 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 11 vertices, video (2021).

Crossrefs

Cf. A010527 (volume of double 5-pyramid), A081314, A081366, A122553 (volume of octahedron), A339259, A339260, A339261, A339262.