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

Author

Clark Kimberling, Feb 09 2009

Keywords

• nonn
• cons

Comments

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

Examples

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

Links

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

Crossrefs

Cf. A195696, A247412.

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

Programs

Maple

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

Mathematica

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

PARI

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

Formula

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