A339262 -id:A339262 - 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).

A340918 Decimal expansion of largest angular separation (in radians) between 10 points on a unit sphere.

Original entry on oeis.org

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

Offset: 1

Hugo Pfoertner, Jan 30 2021

In his habilitation thesis from 1963, Ludwig Danzer provided an interval from 1.1544786 to 1.1544795 (rounded) for this value.

			1.1544798334192707378319618404230211144893004873633425122414214417...

Ludwig Danzer, Endliche Punktmengen auf der 2-Sphaere mit moeglichst grossem Minimalabstand. Habilitationsschrift, Universitaet Goettingen, 1963. See link for the English translation.

Paolo Xausa, Table of n, a(n) for n = 1..10000
Ludwig Danzer, Finite point-sets on S^2 with minimum distance as large as possible, Discrete Mathematics, Volume 60, June-July 1986, Pages 3-66, Table 1, page 63.
Hugo Pfoertner, Visualization of the optimal configuration, (2001).
Teruhisa Sugimoto and Masaharu Tanemura, Exact value of Tammes problem for N=10, arXiv:1509.01768 [math.MG], 12 Sep 2015.

Cf. A080865, A217695, A339262.

First[RealDigits[ArcTan[Sqrt[4/Sqrt[3]*Cos[ArcTan[Sqrt[3*229]/9]/3] + 3]], 10, 100]] (* Paolo Xausa, Feb 27 2025 *)

atan(sqrt((4/sqrt(3))*cos((1/3)*atan(sqrt(3*229)/9))+3))

atan(sqrt((4/sqrt(3))*cos((1/3)*atan(sqrt(3*229)/9)) + 3))

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

Original entry on oeis.org

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

Hugo Pfoertner, Dec 07 2020

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.

			2.35463449506861520323688059263892654160344864269342168599607566...

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).

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

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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A340918 Decimal expansion of largest angular separation (in radians) between 10 points on a unit sphere.

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

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Author

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Crossrefs

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.

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Author

Keywords

Comments

References

Links

Crossrefs