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
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
KEYWORD
nonn,cons
AUTHOR
Clark Kimberling, Feb 09 2009
STATUS
approved