A156547 -id:A156547 - cp's OEIS frontend

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

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

A228496 Decimal expansion of arccos(2/3).

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Aug 23 2013

The value describes the smaller of the internal angles in the triangle of the surfaces of the Tetrakis Hexahedron.
The value equals Pi/2 minus A156547, the 90-degree complement to 41.81031... degrees.
The value is a little bit larger than the 4th root of 1/2, which is 0.8408964... = A228497.
If a ball assimilated to a point rolls without friction on a sphere starting from the top with zero initial velocity, this value is the angle in radians, measured at the center of the sphere, from the top of the sphere to the point at which the ball leaves the surface of the sphere. See Jayanth et al. - Robert FERREOL, Sep 14 2019
The maximum possible value of the least of the nine acute angles between pairs of edges of two randomly disoriented cubes (in radians, see A361618). - Amiram Eldar, Mar 18 2023

			Equals 0.8410686705679302557765... radians  = 48.189685... degrees.

emc2, Bille glissant sur une sphère (in French).
V. Jayanth, C. Raghunandan and Anindya Kumar Biswas, A sphere moving down the surface of a static sphere and a simple phase diagram, arXiv:0808.3531 [physics.class-ph], 2008-2009.
R. J. Mathar, Hierarchical Subdivision of the Simple Cubic Lattice, arXiv preprint arXiv:1309.3705 [math.MG], 2013.
Wikipedia, Tetrakis hexahedron.
Index entries for transcendental numbers

Cf. A156547, A228497, A361618.

Maple
```
evalf(arccos(2/3)) ;
```

RealDigits[ArcCos[2/3], 10, 100][[1]] (* Amiram Eldar, May 24 2021 *)

PARI
```
acos(2/3) \\ Michel Marcus, Sep 14 2019
```

Equals arcsin(sqrt(5)/3).

Equals arctan(sqrt(5)/2). - Amiram Eldar, May 24 2021

A381264 Decimal expansion of the edge length of the dodecahedron inside a circumscribed unit sphere.

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Feb 18 2025

			0.7136441795461798638839396860921757479633721504937...

Paolo Xausa, Table of n, a(n) for n = 0..10000
Wikipedia, Regular Dodecahedron.

Cf. A156547, A001622, A179296.

g := (1+sqrt(5))/2 ; Digits := 100 ; evalf(2/sqrt(3)/g) ;

First[RealDigits[2/(Sqrt[3]*GoldenRatio), 10, 100]] (* Paolo Xausa, Feb 19 2025 *)

Equals 2/(sqrt(3)*A001622) = 1/A179296.
Equals 2*sin(A156547/2).
Minimal polynomial: 9*x^4 - 36*x^2 + 16. - Amiram Eldar, Feb 26 2025

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

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A228496 Decimal expansion of arccos(2/3).

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A381264 Decimal expansion of the edge length of the dodecahedron inside a circumscribed unit sphere.

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