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A156546 Decimal expansion of the central angle of a regular tetrahedron. 10

%I #49 Feb 23 2021 12:20:27

%S 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,

%T 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,

%U 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

%N Decimal expansion of the central angle of a regular tetrahedron.

%C 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).

%C The (minimal) central angle of the other four regular polyhedra are as follows:

%C - cube: A137914,

%C - octahedron: A019669,

%C - dodecahedron: A156547,

%C - icosahedron: A105199.

%C Dihedral angle of two adjacent faces of the octahedron. - _R. J. Mathar_, Mar 24 2012

%C 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

%C Also twice the magic angle (A195696). - _Stanislav Sykora_, Nov 14 2013

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Tetrahedron">Tetrahedron</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Tetrahedral_molecular_geometry">Tetrahedral molecular geometry</a>

%F 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.

%F Two times A195696. - _R. J. Mathar_, Mar 24 2012

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

%t RealDigits[Pi-ArcCos[1/3],10,120][[1]] (* _Harvey P. Dale_, Aug 25 2011 *)

%o (PARI) acos(-1/3) \\ _Charles R Greathouse IV_, Aug 30 2013

%Y Cf. A195696, A247412.

%Y Cf. Platonic solids dihedral angles: A137914 (tetrahedron), A019669 (cube), A236367 (icosahedron), A137218 (dodecahedron). - _Stanislav Sykora_, Jan 23 2014

%K nonn,cons

%O 1,2

%A _Clark Kimberling_, Feb 09 2009

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