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A156546
Decimal expansion of the central angle of a regular tetrahedron.
10
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
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. Platonic solids dihedral angles: A137914 (tetrahedron), A019669 (cube), A236367 (icosahedron), A137218 (dodecahedron). - Stanislav Sykora, Jan 23 2014
Sequence in context: A372948 A102209 A222131 * A154839 A064733 A324998
KEYWORD
nonn,cons
AUTHOR
Clark Kimberling, Feb 09 2009
STATUS
approved