A081314 -id:A081314 - cp's OEIS frontend

A080865 Order of symmetry groups of n points on 3-dimensional sphere with minimal distance between them maximized, also known as hostile neighbor or Tammes problem.

24, 12, 48, 6, 16, 12, 4, 10, 120, 8, 8

Offset: 4

Hugo Pfoertner, Feb 21 2003

If more than one best packing exists (this occurs for n = 15, 62, 76, 117, ...; see Buddenhagen, Kottwitz link) for a given n, the one with the largest symmetry group is chosen. A conjectured (except n=24) continuation of the sequence starting with n=15 would be: 3 16 4 2 2 12 1 1 1 24 3 2 4 1 1 6 5 6 3 2 1 4 2 24 1 3 1 10 1 2 1 2 1 24 2 12.

L. Fejes Toth, Lagerungen in der Ebene auf der Kugel und im Raum, 2nd. ed., Springer-Verlag, Berlin, Heidelberg 1972.

James Buddenhagen and D. A. Kottwitz, Multiplicity and Symmetry Breaking in (Conjectured) Densest Packings of Congruent Circles on a Sphere.
D. A. Kottwitz, The Densest Packing of Equal Circles on a Sphere, Acta Cryst. (1991). A47, 158-165
O. R. Musin and A. S. Tarasov, The strong thirteen spheres problem, Discrete Comput. Geom., 48 (2012), 128-141, arXiv:1002.1439 [math.MG], 2010-2012.
O. R. Musin and A. S. Tarasov, The Tammes problem for N=14, Experimental Mathematics, 24 (2015), 460-468, arXiv:1410.2536 [math.MG].
Hugo Pfoertner, Arrangement of points on a sphere. Visualization of the best known solutions of the Tammes problem.
K. Schuette and B. L. van der Waerden, Auf welcher Kugel haben 5, 6, 7, 8 oder 9 Punkte mit Mindestabstand Eins Platz?, Math. Annalen, 123 (1951), 96-124.
K. Schuette and B. L. van der Waerden, Auf welcher Kugel haben 5, 6, 7, 8 oder 9 Punkte mit Mindestabstand Eins Platz?, Math. Annalen, 123 (1951), 96-124.
N. J. A. Sloane, Library of 3-d packings

A080866 gives the number of shortest edges which make up the rigid framework of the arrangement.
Cf. A081314, A084827, A242088, A242617, A217695, A268487.
Cf. A342559 (point numbers where records of packing density occur).

A081366 Number of distinct edge lengths in the convex hull of the maximal volume arrangements of n points on a sphere.

Original entry on oeis.org

1, 2, 1, 2, 3, 3, 3, 8, 1, 11, 3, 5, 3, 14, 8, 25, 9, 29, 16, 11, 18, 34, 37, 6

Offset: 4

Hugo Pfoertner, Mar 19 2003

			a(8)=3 because the corresponding arrangement has 6 edges of length 1.1383499, 8 edges of length 1.264911.. and 4 edges of length 1.4554505, i.e. 3 distinct edge lengths.

See under A081314.

Hugo Pfoertner, Maximal Volume Arrangements: List of edges.

Symmetry groups of maximal volume arrangements: A081314. Distinct distances for minimal energy configurations: A033177.

A081621 Number of n-node triangulations of the sphere with minimal degree 5.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 3, 4, 12, 23, 73, 192, 651, 2070, 7290, 25381, 91441, 329824, 1204737, 4412031, 16248772, 59995535, 222231424, 825028656, 3069993552, 11446245342, 42758608761, 160012226334, 599822851579, 2252137171764, 8469193859271, 31896058068930

Offset: 4

Hugo Pfoertner, Mar 24 2003

nonn

Other face sizes larger than 5 and 6 are allowed and there can be more than 12 vertices with degree 5.
Convex polytopes with minimum degree at least 5. The sequence is extracted from the file more-counts.txt that comes with the plantri distribution.
Grace conjectured that all polyhedra inscribed in the unit sphere with maximal volume are "medial" (all faces triangular and vertex degree either m or m+1 where m < 6 - 12/n < m+1). For n = 12 and n > 13 the medial polyhedra have 12 vertices of degree 5 and n-12 vertices of degree 6. All known numerical solutions of the maximal volume problem (A081314) have this property.
The triangulated arrangements of points on a sphere with icosahedral symmetry given by Hardin, Sloane and Smith are examples for large n.

			With vertices denoted by letters a, b, ... the neighbor lists are for a(14)=1: (bcdef, afghc, abhid, acije, adjkf, aeklgb, bflmh, bgmic, chmnjd, dinke, ejnlf, fknmg, glnih, imlkj).
a(15)=1: (bcdefg, aghic, abijd, acjke, adklf, aelmg, afmhb, bgmni, bhnjc, cinokd, djole, ekomf, flonhg, hmoji, jnmlk); a(16)=3: (bcdef, afghc, abhijd, acjke, adklf, aelmgb, bfmnh, bgnic, chnoj, ciokd, djople, ekpmf, flpng, gmpoih, inpkj, konml), (bcdef, afghc, abhijd, acjke, adklf, aelmgb, bfmnh, bgnic, chnoj, ciopkd, djple, ekpmf, flpong, gmoih, inmpj, jomlk), (bcdef, afghijc, abjkd, ackle, adlmf, aemgb, bfmnh, bgnoi, bhopj, bipkc, cjpld, dkponme, elngf, gmloh, hnlpi, iolkj).

Gunnar Brinkmann and Brendan McKay, plantri and fullgen programs for generation of certain types of planar graph.
Gunnar Brinkmann and Brendan McKay, plantri and fullgen programs for generation of certain types of planar graph [Cached copy, pdf file only, no active links, with permission]
G. Brinkmann and B. D. McKay, Construction of planar triangulations with minimum degree 5, Discr. Math. 301 (2005), 147-163.
CombOS - Combinatorial Object Server, generate planar graphs.
D. W. Grace, Search for largest polyhedra, Math. Comp. 17, 197-199 (1963).
R. H. Hardin, N. J. A. Sloane and W. D. Smith, Spherical Codes with Icosahedral Symmetry.
Hugo Pfoertner, Icosahedral best coverings.
Sage, Common Graphs.
Thom Sulanke, Generating triangulations of surfaces (surftri), (also subpages).

Cf. A000109, A000103, A081314.

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), May 05 2007

a(41) computed with plantri by Jan Goedgebeur, Dec 03 2021

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

Original entry on oeis.org

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

Hugo Pfoertner, Nov 29 2020

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.

			1.8157161042244203975084949306331777890131009552754398376663729...

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).

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

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

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

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

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

Original entry on oeis.org

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.

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.

cp's OEIS Frontend

A080865 Order of symmetry groups of n points on 3-dimensional sphere with minimal distance between them maximized, also known as hostile neighbor or Tammes problem.

Views

Author

Keywords

Comments

References

Links

Crossrefs

A081366 Number of distinct edge lengths in the convex hull of the maximal volume arrangements of n points on a sphere.

Views

Author

Keywords

Examples

References

Links

Crossrefs

A081621 Number of n-node triangulations of the sphere with minimal degree 5.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

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

Views

Author

Keywords

Comments

Examples

Links

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.

Views

Author

Keywords

Comments

References

Links

Crossrefs