A339263 -id:A339263 - cp's OEIS frontend

A339261 Decimal expansion of the conjecturally maximum possible volume of a polyhedron with 9 vertices inscribed in the unit sphere.

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

Offset: 1

Hugo Pfoertner, Dec 05 2020

			2.0437501158996398411663654642269853338632606152947518187182157956871...

R. H. Hardin, N. J. A. Sloane and W. D. Smith, Maximal Volume Spherical Codes.
Hugo Pfoertner, Visualization of Polyhedron, (1999).
Hugo Pfoertner, 9-Vertex-Polyhedron with maximum volume inscribed in a sphere, YouTube video, Feb 10 2021.

Cf. A010527 (volume of double 5-pyramid), A081314, A081366, A122553 (volume of octahedron), A339259, A339260, A339261, A339262, A339263.

RealDigits[3*Sqrt[2*Sqrt[3] - 3], 10, 120][[1]] (* Amiram Eldar, Jun 28 2023 *)

PARI
```
3*sqrt(2*sqrt(3) - 3)
```

Equals 3*sqrt(2*sqrt(3) - 3).

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

Original entry on oeis.org

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

Hugo Pfoertner, Dec 07 2020

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.

			2.218711131545399403247282751128417013810725374663344381750049084201...

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).

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

A339546 Number of at least 3-connected planar triangulations on n vertices such that the minimum valence of any vertex in the mesh is maximized and the number of vertices with this minimum valence is minimized.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 6, 6, 15, 17, 40, 45, 89, 116, 199, 271

Offset: 4

Hugo Pfoertner, Dec 08 2020

It is conjectured that for a polyhedron with maximum possible volume inscribed in a sphere the stated condition is necessary. All known polyhedra with conjecturally maximum volume satisfy this condition. For n <= 15 the given condition determines a unique mesh topology for each n, which coincides with the proved optimal solutions for n <= 8.

See A081314 for references and links.

Gunnar Brinkmann and Brendan McKay, plantri and fullgen programs for generation of certain types of planar graph.
Hugo Pfoertner, Construction of sequence, table of vertex degrees of triangulations (2020).

Cf. A081314, A081621, A339260, A339261, A339262, A339263.

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A339261 Decimal expansion of the conjecturally maximum possible volume of a polyhedron with 9 vertices inscribed in the unit sphere.

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

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A339546 Number of at least 3-connected planar triangulations on n vertices such that the minimum valence of any vertex in the mesh is maximized and the number of vertices with this minimum valence is minimized.

Views

Author

Keywords

Comments

References

Links

Crossrefs