A137914 -id:A137914 - cp's OEIS frontend

A137915 Decimal expansion of 180*arccos(1/3)/Pi.

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

Offset: 2

Rick L. Shepherd, Feb 22 2008

Dihedral angle in degrees of regular tetrahedron.

Polar angle (or apex angle) of the cone that subtends exactly one third of the full solid angle. - Stanislav Sykora, Feb 20 2014

			70.52877936550930863075400066003756403699315689909205171182889364396025356...

G. C. Greubel, Table of n, a(n) for n = 2..10001
Frank Jackson and Eric W. Weisstein, Tetrahedron
Eric Weisstein's World of Mathematics, Dihedral Angle

Cf. A137914 (same in radians).

SetDefaultRealField(RealField(100)); R:= RealField(); 180*Arccos(1/3)/Pi(R); // G. C. Greubel, Aug 20 2018

RealDigits[180 ArcCos[1/3]/Pi,10,120][[1]] (* Harvey P. Dale, Jul 13 2013 *)

PARI
```
180*acos(1/3)/Pi
```

180*A137914/Pi = 180*arccos(1/3)/Pi.

A195729 Decimal expansion of arctan(3).

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Sep 23 2011

			arctan(3) = 1.2490457723982544258299170772...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Index entries for transcendental numbers

Cf. A105531, A137914, A188615.

SetDefaultRealField(RealField(100)); Arctan(3); // G. C. Greubel, Aug 20 2018

r = 3;
N[ArcTan[r], 100]
RealDigits[%]  (* A195729 *)
N[ArcCot[r], 100]
RealDigits[%]  (* A105531 *)
N[ArcSec[r], 100]
RealDigits[%]  (* A137914 *)
N[ArcCsc[r], 100]
RealDigits[%]  (* A188615 *)

atan(3) \\ Charles R Greathouse IV, Sep 23 2014

Equals arctan(1) + arctan(1/2). - Charles R Greathouse IV, Sep 23 2014

Equals arcsin(3/sqrt(10)) = arccos(sqrt(1/10)). - Amiram Eldar, Jul 11 2023

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.

A202473 Decimal expansion of the surface area of a unit Reuleaux triangle.

Original entry on oeis.org

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

Offset: 1

Eric W. Weisstein, Dec 19 2011

			2.9754717165844016292...

Eric Weisstein's World of Mathematics, Reuleaux Tetrahedron

Cf. A137914, A228824.

RealDigits[8*Pi - 18*ArcCos[1/3], 10, 100][[1]]

8*Pi - 18*acos(1/3) \\ Stefano Spezia, May 26 2025

Equals 8*Pi - 18*arccos(1/3).

A386435 Decimal expansion of the largest dihedral angle, in radians, in a triangular bipyramid (Johnson solid J_12).

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Aug 20 2025

Also the largest dihedral angle in a triangular orthobicupola (Johnson solid J_27) and the second largest dihedral angle in an augmented truncated tetrahedron (Johnson solid J_65).

			2.461918834681549364269858356495974751420680018710...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Wikipedia, Augmented truncated tetrahedron.
Wikipedia, Triangular bipyramid.
Wikipedia, Triangular orthobicupola.

Cf. A137914 (J_12 smallest dihedral angle).

Cf. A020775 (J_12 volume), A104956 (J_12 surface area).

First[RealDigits[ArcCos[-7/9], 10, 100]] (* or *)
First[RealDigits[Max[PolyhedronData["J12", "DihedralAngles"]], 10, 100]]

Equals arccos(-7/9).

A289505 Decimal expansion of arcsec(3)/(2*Pi).

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Jul 07 2017

			0.195913276015303635085427779611215...

Steven R. Finch, Mean width of a regular simplex, arxiv:1111.4976 [math.MG], 2011-2016, S_2.
W. V. Gehrlein, Condorcet's paradox and the Condorcet efficiency of voting rules, Mathematica Japonica 45, 173-199, 1997.

Cf. A086201, A137914.

Maple
```
arcsec(3)/2/Pi ; evalf(%) ;
```

RealDigits[ArcSec[3]/(2 Pi),10,120][[1]] (* Harvey P. Dale, Jul 21 2021 *)

acos(1/3)/(2*Pi) \\ Michel Marcus, Jul 07 2017

from mpmath import mp, asec, pi
mp.dps=89
print([int(z) for z in list(str(asec(3)/(2*pi))[2:-1])]) # Indranil Ghosh, Jul 07 2017

Equals A137914*A086201.
From Robert FERREOL, Mar 21 2018: (Start)
Equals arctan(2*sqrt(2))/(2*Pi).
Equals (1/(2*Pi))*Integral_{t>=sqrt(2)/4} 1/(1+t^2).
Equals Probability(X>sqrt(2)/4)/2, if X is a Cauchy distributed random variable of location parameter 0 and scale parameter 1.
Equals the asymptotic probability p that A is predominantly preferred to B and B predominantly preferred to C when n persons provide a preference list of three candidates A, B, C (with a uniform distribution on voter preferences); the asymptotic probability that A > B > C > A or A > C > B > A (where ">" means "predominantly preferred to") is 3p-1/2 = 8.77...% (Condorcet paradox); the contrary probability (existence of a Condorcet winner) is 3/2-3p = 91.226...%.
See Gehrlein link. (End)

A349605 Decimal expansion of the probability that the intersection of a cube with random plane that passes through its center is a hexagon.

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Nov 23 2021

The normal to the random plane is in the direction from the center of the cube to a point uniformly chosen at random on the surface of a sphere whose center coincides with the center of the cube.

			0.35095931218364362102513335533458546789977189663640...

Su Pernu Mero, Problem 2065, Mathematics Magazine, Vol. 92, No. 1, (2019), p. 73; Solution, by Bill Cowieson, ibid., Vol. 93, No. 1 (2020), pp. 76-77.
Yawen Zhang, A purely 3-D geometrical solution to Mathematics Magazine Problem 2065, arXiv:2010.12349 [math.HO], 2020.

Cf. A137914, A188615.

RealDigits[1 - 6 * ArcSin[1/3] / Pi, 10, 100][[1]]

1 - 6*asin(1/3)/Pi \\ Michel Marcus, Nov 23 2021

Equals 1 - 6*arcsin(1/3)/Pi.

Equals 6*arccos(1/3)/Pi - 2.

cp's OEIS Frontend

A137915 Decimal expansion of 180*arccos(1/3)/Pi.

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A195729 Decimal expansion of arctan(3).

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

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

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A202473 Decimal expansion of the surface area of a unit Reuleaux triangle.

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Mathematica

PARI

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A386435 Decimal expansion of the largest dihedral angle, in radians, in a triangular bipyramid (Johnson solid J_12).

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Mathematica

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A289505 Decimal expansion of arcsec(3)/(2*Pi).

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