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

1,2

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

Links

Table of n, a(n) for n=1..105.

Wikipedia, Tetrahedron

Wikipedia, Tetrahedral molecular geometry

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.

Two times A195696. - R. J. Mathar, Mar 24 2012

Example

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

Mathematica

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

Prog

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

Crossrefs

Cf. A195696, A247412.

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

Sequence in context: A227817 A102209 A222131 * A154839 A064733 A324998

Adjacent sequences: A156543 A156544 A156545 * A156547 A156548 A156549

Keyword

nonn,cons

Author

Clark Kimberling, Feb 09 2009

Status

approved