A217695 -id:A217695 - 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).

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))

A383859 Central angle of the solution of the Tammes problem for 7 points on the sphere (in radians).

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, May 12 2025

			1.3590798976326601418850028816473327537..

Laslo Hars, Numerical solutions of the Tammes problem for up to 60 points, Nov 2020, N=7.
K. Schutte and B. L. van der Waerden, Auf welcher Kugel haben 5, 6, 7, 8 oder 9 Punkte mit Mindestabstand eins Platz?, Math. Ann. 123 (1951) 96-124.
Wikipedia, Tammes problem

Cf. A019819, A019669 (N=6), A381756 (N=8), A137914 (N=9), A340918 (N=10), A105199 (N=11 and N=12). A217695 (N=13), A383860 (N=14), A383861 (N=24).

cos(4*Pi/9) ; %/(1-%) ; arccos(%) ; evalf(%,120) ;

cos( this ) = cos phi/(1- cos phi) where cos(phi)=A019819.

A383860 Central angle of the solution of the Tammes problem for 14 points on the sphere (in radians).

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, May 12 2025

			0.971634742886224075941694947628...

Laslo Hars, Numerical solutions of the Tammes problem for up to 60 points, Nov 2020, N=14.
O. R. Musin and A. S. Tarasov, The Tammes problem for N=14, Exp. Math. 24 (2015) 460-468.
Wikipedia, Tammes problem

Cf. A019669 (N=6), A383859 (N=7), A381756 (N=8), A137914 (N=9), A340918 (N=10), A105199 (N=11 and N=12). A217695 (N=13), A383861 (N=24).

Digits := 120 ;
g := proc(c,x)
    2*arccot(c*tan(x/2)) ;
end proc:
f := proc(x)
    local c,x1,x2,x3,x4,x5 ;
    c := cos(x)/(1-cos(x)) ;
    x1 := Pi-x ;
    x2 := g(c,x1) ;
    x3 := 2*Pi-2*x-x2 ;
    x4 := g(c,x3) ;
    x5 := 2*Pi-x-2*x4 ;
    2*Pi-2*x-x3-g(c,x5) ;
end proc:
x := 1.2 ; y := 1.21 ;
for i from 1 to 500 do
    z := (x+y)/2 ;
    if f(z) > 0. then
        x := z ;
    else
        y := z ;
    end if;
    cos(z)/(1-cos(z)) ;
    if modp(i,20) =0 then
        arccos(%) ; evalf(%,120) ; print(%) ;
    end if;
    if x > y then
        break ;
    end if;
end do:

A383861 Central angle of the solution of the Tammes problem for 24 points on the sphere (in radians).

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, May 12 2025

			0.762547738750982556743106092114...

Laslo Hars, Numerical solutions of the Tammes problem for up to 60 points, Nov 2020, N=24.
R. M. Robinson, Arrangements of 24 points on the sphere, Math. Ann. 144 (1961) 17-48.
Wikipedia, Tammes problem

Cf. A058265, A019669 (N=6), A383859 (N=7), A381756 (N=8), A137914 (N=9), A340918 (N=10), A105199 (N=11 and N=12). A217695 (N=13), A383860 (N=14).

t := (1+(19+3*sqrt(33))^(1/3)+(19-3*sqrt(33))^(1/3))/3 ; arccos((t-1)/(3-t)) ; evalf(%,120);

cos( this ) = (t-1)/(3-t) where t=A058265.

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.

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

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A383859 Central angle of the solution of the Tammes problem for 7 points on the sphere (in radians).

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Maple

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A383860 Central angle of the solution of the Tammes problem for 14 points on the sphere (in radians).

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Maple

A383861 Central angle of the solution of the Tammes problem for 24 points on the sphere (in radians).

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Maple

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