A156546 -id:A156546 - cp's OEIS frontend

A137914 Decimal expansion of arccos(1/3).

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

A195696 Decimal expansion of arccos(sqrt(1/3)) and of arcsin(sqrt(2/3)) and arctan(sqrt(2)).

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Sep 23 2011

Angle (in radians) between an edge and (the normal of) a face of the regular tetrahedron. - R. J. Mathar, Feb 23 2012
Also known as magic angle; root of P_2(cos(theta)), with P_2(x) being second-order Legendre polynomial. - Stanislav Sykora, May 25 2012
From Stanislav Sykora, Nov 14 2013: (Start)
Also the angle between the body diagonal of a cube and an incident edge, and therefore the polar angle of the cone circumscribed to a cube from one of its vertices.
Also half of the tetrahedral angle (A156546).
In nuclear magnetic resonance, the angle, with respect to the direction of the main magnetic field, under which a solid sample needs to be spun in order to average to zero unwanted dipole-dipole spin interactions (the magic angle spinning, or MAS, technique). (End)
Also <3_2> in Conway et al. (1999). - Eric W. Weisstein, Nov 06 2024

			0.9553166181245092781638571025157577... (= 54.73561031... degrees).

G. C. Greubel, Table of n, a(n) for n = 0..10000
John H. Conway, Charles Radin, and Lorenzo Sadun, On Angles Whose Squared Trigonometric Functions are Rational, arXiv:math-ph/9812019, 1998. See also Discr. Computat. Geom. (1999) Vol. 22, 321-332.
H. B. Dwight, Tables of Integrals and other Mathematical Data. 507.13, 507.22 in Inverse trigonometric functions. New York: Macmillan Publishing, p. 120, 1961.
Steven R. Finch, Errata and Addenda to Mathematical Constants, arXiv:2001.00578 [math.HO], 2020-2022; p. 53.
C. O. Horgan and J. G. Murphy, On an angle with magical properties, Notices Amer. Math. Soc., 69:1 (2022), 22-25.
G. Jacob Martens, Rational right triangles and the Congruent Number Problem, arXiv:2112.09553 [math.GM], 2021, see section 2.6.2 The Trinity vectors, the magic angle, equation (34).
Stanislav Sýkora, Magnetic Resonance on OEIS, Stan's NMR Blog (Dec 31, 2014), Retrieved Nov 12, 2019.
Eric Weisstein's World of Mathematics, Dehn Invariant.
Wikipedia, Tetrahedron
Wikipedia, Magic angle
Index entries for transcendental numbers

Cf. A156546, A195695, A197739, A210974 (in degrees), A243445.

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

RealDigits[ArcTan[Sqrt[2]],10,120][[1]] (* Harvey P. Dale, Dec 13 2014 *)

atan(sqrt(2)) \\ G. C. Greubel, Jul 05 2017

Equals i*log(sqrt(1/3) - i*sqrt(2/3)). - Andrea Pinos, Nov 03 2023

Equals A156546/2 = 2*A197739. - Hugo Pfoertner, Nov 06 2024

A137218 Decimal expansion of the argument of -1 + 2*i.

Original entry on oeis.org

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

Offset: 1

Matt Rieckman (mjr162006(AT)yahoo.com), Mar 06 2008

Gives closed forms for many arctangent values:
arctan(2) = Pi - a, arctan(1/2) = a - Pi/2,
arctan(3) = a - Pi/4, arctan(1/3) = 3*Pi/4 - a,
arctan(7) = 7*Pi/4 - 2*a, arctan(1/7) = 2*a - 5*Pi/4,
arctan(4/3) = 2*a - Pi and arctan(3/4) = 3*Pi/2 - 2*a.
Dihedral angle in the dodecahedron (radians). - R. J. Mathar, Mar 24 2012
Larger interior angle (in radians) of a golden rhombus; A105199 is the smaller interior angle. - Eric W. Weisstein, Dec 17 2018

			2.0344439357957027354455779231...

Rick L. Shepherd, Table of n, a(n) for n = 1..20000
Eric Weisstein's World of Mathematics, Golden Rhombus
Wikipedia, Dodecahedron
Index entries for transcendental numbers.

Platonic solids' dihedral angles: A137914 (tetrahedron), A156546 (octahedron), A019669 (cube), A236367 (icosahedron). - Stanislav Sykora, Jan 23 2014
Cf. A242723 (same in degrees).
Cf. A105199 (smaller interior angle of the golden rhombus).

RealDigits[Pi - ArcTan[2], 10, 120][[1]] (* Harvey P. Dale, Aug 08 2014 *)

default(realprecision, 120);
acos(-1/sqrt(5)) \\ or
arg(-1+2*I) \\ Rick L. Shepherd, Jan 26 2014

Equals Pi - arctan(2) = A000796 - A105199 = 2*A195723.

Corrected a typo in the sequence Matt Rieckman (mjr162006(AT)yahoo.com), Feb 05 2010

More terms from Rick L. Shepherd, Jan 26 2014

A236367 Dihedral angle in a regular icosahedron (radians).

Original entry on oeis.org

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

Offset: 1

Stanislav Sykora, Jan 23 2014

			2.41186499736282687500784672346618218880066348532739213...

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

Cf. A001622, Platonic solids dihedral angles: A137914 (tetrahedron), A156546 (octahedron), A019669 (cube), A137218 (dodecahedron).

RealDigits[2 * ArcTan[GoldenRatio^2], 10, 120][[1]] (* Amiram Eldar, May 17 2023 *)

PARI
```
2*atan((3+sqrt(5))/2)
```

Equals 2*arctan(phi^2) = 2*arctan(A001622^2) = 2*arctan((3+sqrt(5))/2).

A378208 Decimal expansion of the dihedral angle, in radians, between any two adjacent faces in a triakis tetrahedron.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Nov 21 2024

The triakis tetrahedron is the dual polyhedron of the truncated tetrahedron.

			2.2605713275803962793413578116086559655552841805381...

Wikipedia, Triakis tetrahedron.
Index entries for transcendental numbers.

Cf. A378204 (surface area), A378205 (volume), A378206 (inradius), A378207 (midradius).

Cf. A137914 and A156546 (dihedral angles of a truncated tetrahedron).

First[RealDigits[ArcCos[-7/11], 10, 100]] (* or *)
First[RealDigits[First[PolyhedronData["TriakisTetrahedron", "DihedralAngles"]], 10, 100]]

Equals arccos(-7/11).

A363437 Decimal expansion of the volume of the regular tetrahedron inscribed in the unit-radius sphere.

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Jun 02 2023

			0.51320023927966734623035447155729551613120155668455...

Eric Weisstein's World of Mathematics, Regular Tetrahedron.
Wikipedia, Tetrahedron: Regular tetrahedron.
Index entries for algebraic numbers, degree 2.

Cf. A118273 (cube), A122553 (regular octahedron), A339259 (regular icosahedron), A363438 (regular dodecahedron).

Other constants related to the regular tetrahedron: A020781, A020829, A137914, A156546, A187110, A210974, A232812, A236555.

Mathematica
```
RealDigits[8/(9*Sqrt[3]), 10, 120][[1]]
```

8/9/sqrt(3) \\ Charles R Greathouse IV, Feb 07 2025

Equals 8/(9*sqrt(3)).
Equals A118273 / 3.
Equals A020829 / A187110 ^ 3.

A156547 Decimal expansion of the central angle of a regular dodecahedron.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Feb 09 2009

If A and B are neighboring vertices of a regular dodecahedron having center O, then the central angle AOB is this number; the exact value is arccos((1/3)*sqrt(5)) = arcsin(2/3).
The (minimal) central angle of the other four regular polyhedra are as follows:
- tetrahedron: A156546,
- cube: A137914,
- octahedron: A019669,
- icosahedron: A105199.

			arccos((1/3)*sqrt(5))=0.729727656226966..., or, in degrees,
41.810314895778598065857916730578259531014119535901347753...

Cf. A001622.

Cf. A019669, A105199, A137914, A156546.

evalf(arcsin(2/3)); #  Robert FERREOL, Sep 14 2019

RealDigits[ArcCos[Sqrt[5]/3],10,120][[1]] (* Harvey P. Dale, Feb 23 2015 *)

asin(2/3) \\ Charles R Greathouse IV, May 28 2013

The dodecahedron has 12 faces and 20 vertices. To find the central angle, we need any neighboring pair of vertices. Here are all 20 vertices:
- (d,d,d) where d is 1 or -1 (that's 8 vertices);
- (0, d*(t-1),d*t), where d is 1 or -1 and d = golden ratio = (1+sqrt(5))/2;
- (d*(t-1), d*t, 0); and ((d*t,0,d*(t-1)).
An example of a neighboring pair is (1,1,1) and (0,t,t-1).
Apply the usual formula for the cosine of the angle between two vectors.
Equals 2 * arccot(phi^2), where phi is the golden ratio (A001622). - Amiram Eldar, Jul 06 2023

A247412 Decimal expansion of the tetrahedral angle (in degrees).

Original entry on oeis.org

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

Offset: 3

Stanislav Sykora, Sep 16 2014

Angle between chemical bonds in the sp^3 hybridization scheme.
Angle between any two non-parallel bonds in a diamond crystal.
For its value in radians and further comments, see A156546.

			109.4712206344906913692459993399624359630068431009079482881711063560...

Stanislav Sykora, Table of n, a(n) for n = 3..2000

Cf. A156546.

RealDigits[N[ArcCos[-1/3]*180/Pi, 100]][[1]] (* Vaclav Kotesovec, Sep 17 2014 *)

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

A387148 Decimal expansion of the largest dihedral angle, in radians, in a gyroelongated square pyramid (Johnson solid J_10).

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Aug 19 2025

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

Also the analogous dihedral angle in a gyroelongated square bipyramid (Johnson solid J_17).

			2.76759950111674795948641922563774170695233176699...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Wikipedia, Gyroelongated square bipyramid.
Wikipedia, Gyroelongated square pyramid.

Cf. other J_10 dihedral angles: A156546, A387149, A387150.
Cf. A179638 (J_10 volume), A374948 (J_10 surface area).
Cf. A002193, A011027.

First[RealDigits[ArcCos[(1 - Sqrt[2] - 32^(1/4))/3], 10, 100]] (* or *)
First[RealDigits[Max[PolyhedronData["J10", "DihedralAngles"]], 10, 100]]

Equals arccos((1 - sqrt(2) - 2*2^(1/4))/3) = arccos((1 - A002193 - A011027)/3).

A387149 Decimal expansion of the second largest dihedral angle, in radians, in a gyroelongated square pyramid (Johnson solid J_10).

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Aug 19 2025

This is the dihedral angle between adjacent triangular faces in the square antiprism part of the solid.

Also the analogous dihedral angle in a gyroelongated square bipyramid (Johnson solid J_17).

			2.2261954369024298095991888315497640747046712455090...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Wikipedia, Gyroelongated square bipyramid.
Wikipedia, Gyroelongated square pyramid.

Cf. other J_10 dihedral angles: A156546, A387148, A387150.
Cf. A179638 (J_10 volume), A374948 (J_10 surface area).
Cf. A010466.

First[RealDigits[ArcCos[(1 - Sqrt[8])/3], 10, 100]] (* or *)
First[RealDigits[RankedMax[Union[PolyhedronData["J10", "DihedralAngles"]], 2], 10, 100]]

Equals arccos((1 - 2*sqrt(2))/3) = arccos((1 - A010466)/3).

cp's OEIS Frontend

A137914 Decimal expansion of arccos(1/3).

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A195696 Decimal expansion of arccos(sqrt(1/3)) and of arcsin(sqrt(2/3)) and arctan(sqrt(2)).

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A137218 Decimal expansion of the argument of -1 + 2*i.

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A236367 Dihedral angle in a regular icosahedron (radians).

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A378208 Decimal expansion of the dihedral angle, in radians, between any two adjacent faces in a triakis tetrahedron.

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A363437 Decimal expansion of the volume of the regular tetrahedron inscribed in the unit-radius sphere.

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A156547 Decimal expansion of the central angle of a regular dodecahedron.

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