A210974 -id:A210974 - cp's OEIS frontend

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

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

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.

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

Original entry on oeis.org

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)).

cp's OEIS Frontend

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

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

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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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