A195696 -id:A195696 - cp's OEIS frontend

A273621 Decimal expansion of the solid angle (in steradians) subtended by a cone having the 'magic' angle A195696 as its polar angle.

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

Offset: 1

Stanislav Sykora, Aug 15 2016

An example of such a cone is the one circumscribed to a cube from one of its vertices. When expressed as a fraction of the full solid angle, this constant leads to A156309.

			2.65558657871115077573713025127469430382620630256473049081011931...

Stanislav Sykora, Table of n, a(n) for n = 1..2000

Cf. A000796, A156309, A195696, A210974.

First@RealDigits@N[2*Pi*(1 - Sqrt[1/3]), 25] (* G. C. Greubel, Aug 15 2016 *)

PARI
```
2*Pi*(1-sqrt(1/3))
```

Equals 2*Pi*(1-sqrt(1/3)) = 4*Pi*A156309 = 2*Pi*(1-cos(A210974)).

A020760 Decimal expansion of 1/sqrt(3).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

If the sides of a triangle form an arithmetic progression in the ratio 1:1+d:1+2d then when d=1/sqrt(3) it uniquely maximizes the area of the triangle. This triangle has approximate internal angles 25.588 degs, 42.941 degs, 111.471 degs. - Frank M Jackson, Jun 15 2011
When a cylinder is completely enclosed by a sphere, it occupies a fraction f of the sphere volume. The value of f has a trivial lower bound of 0, and an upper bound which is this constant. It is achieved iff the cylinder diameter is sqrt(2) times its height, and the sphere is circumscribed to it. A similar constant can be associated with any n-dimensional geometric shape. For 3D cuboids it is A165952. - Stanislav Sykora, Mar 07 2016
The ratio between the thickness and diameter of a dynamically fair coin having an equal probability, 1/3, of landing on each of its two faces and on its side after being tossed in the air. The calculation is based on the dynamic of rigid body (Yong and Mahadevan, 2011). See A020765 for a simplified geometrical solution. - Amiram Eldar, Sep 01 2020
The coefficient of variation (relative standard deviation) of natural numbers: Limit_{n->oo} sqrt((n-1)/(3*n+3)) = 1/sqrt(3). - Michal Paulovic, Mar 21 2023

			0.577350269189625764509148780501957455647601751270126876018602326....

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Sections 8.4.3 and 8.17, pp. 495, 531.

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Ee Hou Yong and L. Mahadevan, Probability, geometry, and dynamics in the toss of a thick coin, American Journal of Physics, Vol. 79, No. 12 (2011), pp. 1195-1201, preprint, arXiv:1008.4559 [physics.class-ph], 2010-2011.
Index entries for algebraic numbers, degree 2.

Cf. A002194 (sqrt(3)), A010701 (1/3).

Cf. A002193, A047235, A165952, A195696, A040001.

evalf(1/sqrt(3)); # Michal Paulovic, Mar 21 2023

RealDigits[N[1/Sqrt[3],200]] (* Vladimir Joseph Stephan Orlovsky, May 27 2010 *)

\\ Works in v2.15.0; n = 100 decimal places
my(n=100); digits(floor(10^n/quadgen(12))) \\ Michal Paulovic, Mar 21 2023

Equals 1/A002194. - Michel Marcus, Oct 12 2014
Equals cosine of the magic angle: cos(A195696). - Stanislav Sykora, Mar 07 2016
Equals square root of A010701. - Michel Marcus, Mar 07 2016
Equals 1 + Sum_{k>=0} -(4*k+1)^(-1/2) + (4*k+3)^(-1/2) + (4*k+5)^(-1/2) - (4*k+7)^(-1/2). - Gerry Martens, Nov 22 2022
Equals (1/2)*(2 - zeta(1/2,1/4) + zeta(1/2,3/4) + zeta(1/2,5/4) - zeta(1/2,7/4)). - Gerry Martens, Nov 22 2022
Has periodic continued fraction expansion [0, 1; 1, 2] (A040001). - Michal Paulovic, Mar 21 2023
Equals Product_{k>=1} (1 + (-1)^k/A047235(k)). - Amiram Eldar, Nov 22 2024
Equals tan(Pi/6) = (1/2)/A010527. - R. J. Mathar, Aug 31 2025

A137914 Decimal expansion of arccos(1/3).

Original entry on oeis.org

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

Offset: 1

Rick L. Shepherd, Feb 22 2008

Dihedral angle in radians of regular tetrahedron.
Arccos(1/3) is the central angle of a cube, made by the center and two neighboring vertices. - Clark Kimberling, Feb 10 2009
Also the complementary tetrahedral angle, Pi-A156546, and therefore related to the magic angle (Pi-2*A195696). - Stanislav Sykora, Jan 23 2014
Polar angle (or apex angle) of the cone that subtends exactly one third of the full solid angle. - Stanislav Sykora, Feb 20 2014
Also the acute angle in the rhombi and isosceles trapezoids in the trapezo-rhombic dodecahedron. - Eric W. Weisstein, Jan 09 2019
Also the angle between the tangent lines to the curves y = sin(x) at y = cos(x) at the points of intersection. - David Radcliffe, Jan 17 2023

			1.2309594173407746821349291782479873757103400093550948390555483336639923144...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Steven R. Finch, Errata and Addenda to Mathematical Constants, p. 58.
Eric Weisstein's World of Mathematics, Tetrahedron.
Eric Weisstein's World of Mathematics, Dihedral Angle.
Eric Weisstein's World of Mathematics, Trapezo-Rhombic Dodecahedron.
Index entries for transcendental numbers

Cf. A137915 (same in degrees), A019670, A195695, A195696, A238238, Platonic solids dihedral angles: A156546 (octahedron), A019669 (cube), A236367 (icosahedron), A137218 (dodecahedron).

SetDefaultRealField(RealField(100)); Arccos(1/3); // G. C. Greubel, Aug 20 2018

RealDigits[ArcCos[1/3], 10, 120][[1]] (* Harvey P. Dale, Jul 06 2018 *)
RealDigits[ArcSec[3], 10, 120][[1]] (* Eric W. Weisstein, Jan 09 2019 *)

PARI
```
acos(1/3)
```

arccos(1/3) = arctan(2*sqrt(2)) = 2*arcsin(sqrt(3)/3) = arcsin(2*sqrt(2)/3).
Equals sqrt(2)*Sum_{k>=0} (-1)^k/(2^k*(2*k+1)). - Davide Rotondo, Jun 07 2025
Equals 2*A195695. - Hugo Pfoertner, Jun 07 2025

A156546 Decimal expansion of the central angle of a regular tetrahedron.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Feb 09 2009

If O is the center of a regular tetrahedron ABCD, then the central angle AOB is this number; exact value is Pi - arccos(1/3).
The (minimal) central angle of the other four regular polyhedra are as follows:
- cube: A137914,
- octahedron: A019669,
- dodecahedron: A156547,
- icosahedron: A105199.
Dihedral angle of two adjacent faces of the octahedron. - R. J. Mathar, Mar 24 2012
Best known as "tetrahedral angle" theta (e.g., in chemistry). Its Pi complement (i.e., Pi - theta) is the dihedral angle between adjacent faces in regular tetrahedron. - Stanislav Sykora, May 31 2012
Also twice the magic angle (A195696). - Stanislav Sykora, Nov 14 2013

			Pi - arccos(1/3) = 1.910633236249018556..., or, in degrees, 109.471220634490691369245999339962435963006843100... = A247412

Wikipedia, Tetrahedron
Wikipedia, Tetrahedral molecular geometry
Index entries for transcendental numbers.

Cf. A195696, A247412.

Cf. Platonic solids dihedral angles: A137914 (tetrahedron), A019669 (cube), A236367 (icosahedron), A137218 (dodecahedron). - Stanislav Sykora, Jan 23 2014

evalf(arccos(-1/3)) ; # R. J. Mathar, Feb 18 2025

RealDigits[Pi-ArcCos[1/3],10,120][[1]] (* Harvey P. Dale, Aug 25 2011 *)

acos(-1/3) \\ Charles R Greathouse IV, Aug 30 2013

Start with vertices (1,1,1), (1,-1,-1,), (-1,1,-1), and (1,-1,1) and apply the formula for cosine of the angle between two vectors.
Equals 2* A195696. - R. J. Mathar, Mar 24 2012
Equals A000796 - A137914 = A247412 / A072097 - R. J. Mathar, Feb 18 2025

A195695 Decimal expansion of arcsin(sqrt(1/3)) and of arccos(sqrt(2/3)).

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Sep 23 2011

The complementary magic angle, that is, Pi/2 - A195696. The angle between the body-diagonal and a congruent face-diagonal of a cube. And also the polar angle of the cone circumscribed to a regular tetrahedron from one of its vertices. - Stanislav Sykora, Nov 21 2013

This is the value of the angle of the circular cone to the axis, that maximizes the volume of the cone enclosed by a given area. See the +plus link. - Michel Marcus, Aug 27 2017

			arcsin(sqrt(1/3)) = 0.61547970867038734106746458912399...

G. C. Greubel, Table of n, a(n) for n = 0..5000
John D. Barrow, Outer space: Archimedean ice cream cones, +plus magazine.
Wikipedia, Polyhedron, and further links therein.
Index entries for transcendental numbers

Cf. A195696 (magic angle), A195698, A020760, A157697, A243445.

[Arcsin(Sqrt(1/3))]; // G. C. Greubel, Nov 18 2017

r = Sqrt[1/3];
N[ArcSin[r], 100]
RealDigits[%]  (* A195695 *)
N[ArcCos[r], 100]
RealDigits[%]  (* A195696 *)
N[ArcTan[r], 100]
RealDigits[%]  (* A019673 *)
N[ArcCos[-r], 100]
RealDigits[%]  (* A195698 *)

atan(1/sqrt(2)) \\ Michel Marcus, Aug 27 2017

Also equals arctan(1/sqrt(2)). - Michel Marcus, Aug 27 2017

A195701 Decimal expansion of arctan(sqrt(2/3)).

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Sep 23 2011

			arctan(sqrt(2/3)) = 0.68471920300...

G. C. Greubel, Table of n, a(n) for n = 0..5000
Index entries for transcendental numbers

Cf. A195695, A195695, A195702.

[Arctan(Sqrt(2/3))]; // G. C. Greubel, Nov 18 2017

r = Sqrt[2/3];
N[ArcSin[r], 100]
RealDigits[%]  (* A195696 *)
N[ArcCos[r], 100]
RealDigits[%]  (* A195695 *)
N[ArcTan[r], 100]
RealDigits[%]  (* A195701 *)
N[ArcCos[-r], 100]
RealDigits[%]  (* A195702 *)

atan(sqrt(2/3)) \\ G. C. Greubel, Nov 18 2017

Equals arcsin(sqrt(2/5)) = arccos(sqrt(3/5)). - Amiram Eldar, Jul 10 2023

A195702 Decimal expansion of arccos(-sqrt(2/3)).

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Sep 23 2011

			arccos(-sqrt(2/3)) = 2.5261129449405...

G. C. Greubel, Table of n, a(n) for n = 1..5000
Index entries for transcendental numbers

Cf. A195701.

[Arccos(-Sqrt(2/3))]; // G. C. Greubel, Nov 18 2017

r = Sqrt[2/3];
N[ArcSin[r], 100]
RealDigits[%]  (* A195696 *)
N[ArcCos[r], 100]
RealDigits[%]  (* A195695 *)
N[ArcTan[r], 100]
RealDigits[%]  (* A195701 *)
N[ArcCos[-r], 100]
RealDigits[%]  (* A195702 *)
RealDigits[ArcCos[-Sqrt[(2/3)]],10,120][[1]] (* Harvey P. Dale, Jan 15 2013 *)

acos(-sqrt(2/3)) \\ G. C. Greubel, Nov 18 2017

Equals Pi - arcsin(sqrt(1/3)) = Pi - arctan(sqrt(1/2)). - Amiram Eldar, Jul 10 2023

A243445 Decimal expansion of the polar angle of the cone circumscribed to a regular dodecahedron from one of its vertices.

Original entry on oeis.org

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

Offset: 1

Stanislav Sykora, Jun 06 2014

The angle is in radians.

			1.20593249868141343750392336173309109440033174266369606513299755...

Stanislav Sykora, Table of n, a(n) for n = 1..2000
Wikipedia, Dodecahedron (use the point coordinates to derive the formula).

Cf. A001622 (phi), A003881 (octahedron), A195695 (tetrahedron), A195696 (cube), A195723 (isosahedron).

RealDigits[ArcCos[1/(GoldenRatio Sqrt[3])],10,120][[1]] (* Harvey P. Dale, May 17 2016 *)

PARI
```
acos(2/(1+sqrt(5))/sqrt(3))
```

arccos(1/(phi*sqrt(3))), where phi = A001622.

arctan(phi^2), where phi = A001622. - Jon Maiga, Nov 11 2018

A377277 Decimal expansion of 12*arctan(sqrt(2)).

Original entry on oeis.org

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

Offset: 2

Paolo Xausa, Oct 23 2024

Dehn invariant of a truncated tetrahedron with unit edge and (negated) of a regular tetrahedron with unit edge.

			11.463799417494111337966285230189093050920976340...

Eric Weisstein's World of Mathematics, Dehn Invariant.
Eric Weisstein's World of Mathematics, Regular Tetrahedron.
Eric Weisstein's World of Mathematics, Truncated Tetrahedron.
Index entries for transcendental numbers

Cf. A195696, A377296.

First[RealDigits[12*ArcTan[Sqrt[2]], 10, 100]] (* or *)
First[RealDigits[PolyhedronData["Tetrahedron", "DehnInvariant"], 10, 100]]

12*atan(sqrt(2)) \\ Charles R Greathouse IV, Nov 20 2024

Equals 12*A195696 = A377296/2.

A387295 Decimal expansion of the second largest dihedral angle, in radians, in a gyroelongated triangular cupola (Johnson solid J_22).

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Aug 25 2025

This is the dihedral angle between a triangular face in the antiprism part of the solid and a square face in the cupola part of the solid.

Also the analogous dihedral angle in a gyroelongated triangular bicupola (Johnson solid J_44).

			2.6814372804191827475908005056128080315848833860639...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Wikipedia, Gyroelongated triangular bicupola.
Wikipedia, Gyroelongated triangular cupola.

Cf. other J_22 dihedral angles: A195698, A387294, A387296, A387297.
Cf. A344076 (J_22 volume), A344077 (J_22 surface area).
Cf. A385256 (J_44 volume), A385257 (J_44 surface area).
Cf. A195696, A246724.

First[RealDigits[ArcTan[Sqrt[2]] + ArcCos[1 - Sqrt[12]/3], 10, 100]] (* or *)
First[RealDigits[RankedMax[Union[PolyhedronData["J22", "DihedralAngles"]], 2], 10, 100]]

Equals arccos(sqrt(3)/3) + arccos(1 - 2*sqrt(3)/3) = A195696 + arccos(-A246724).

cp's OEIS Frontend

A273621 Decimal expansion of the solid angle (in steradians) subtended by a cone having the 'magic' angle A195696 as its polar angle.

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A020760 Decimal expansion of 1/sqrt(3).

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A137914 Decimal expansion of arccos(1/3).

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A156546 Decimal expansion of the central angle of a regular tetrahedron.

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A195695 Decimal expansion of arcsin(sqrt(1/3)) and of arccos(sqrt(2/3)).

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Magma

Mathematica

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A195701 Decimal expansion of arctan(sqrt(2/3)).

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A195702 Decimal expansion of arccos(-sqrt(2/3)).

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