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A156546 Decimal expansion of the central angle of a regular tetrahedron. 6
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 (list; constant; graph; refs; listen; history; text; internal format)
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 A020841

Adjacent sequences:  A156543 A156544 A156545 * A156547 A156548 A156549

KEYWORD

nonn,cons

AUTHOR

Clark Kimberling, Feb 09 2009

STATUS

approved

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Last modified December 21 20:27 EST 2014. Contains 252326 sequences.