cp's OEIS Frontend

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

%I A156546 #57 Mar 06 2025 08:22:34
%S A156546 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 A156546 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 A156546 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 A156546 Decimal expansion of the central angle of a regular tetrahedron.
%C A156546 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 A156546 The (minimal) central angle of the other four regular polyhedra are as follows:
%C A156546 - cube: A137914,
%C A156546 - octahedron: A019669,
%C A156546 - dodecahedron: A156547,
%C A156546 - icosahedron: A105199.
%C A156546 Dihedral angle of two adjacent faces of the octahedron. - _R. J. Mathar_, Mar 24 2012
%C A156546 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 A156546 Also twice the magic angle (A195696). - _Stanislav Sykora_, Nov 14 2013
%H A156546 Wikipedia, <a href="http://en.wikipedia.org/wiki/Tetrahedron">Tetrahedron</a>
%H A156546 Wikipedia, <a href="https://en.wikipedia.org/wiki/Tetrahedral_molecular_geometry">Tetrahedral molecular geometry</a>
%H A156546 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%F A156546 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 A156546 Equals 2* A195696. - _R. J. Mathar_, Mar 24 2012
%F A156546 Equals A000796 - A137914 = A247412 / A072097 - _R. J. Mathar_, Feb 18 2025
%e A156546 Pi - arccos(1/3) = 1.910633236249018556..., or, in degrees, 109.471220634490691369245999339962435963006843100... = A247412
%p A156546 evalf(arccos(-1/3)) ; # _R. J. Mathar_, Feb 18 2025
%t A156546 RealDigits[Pi-ArcCos[1/3],10,120][[1]] (* _Harvey P. Dale_, Aug 25 2011 *)
%o A156546 (PARI) acos(-1/3) \\ _Charles R Greathouse IV_, Aug 30 2013
%Y A156546 Cf. A195696, A247412.
%Y A156546 Cf. Platonic solids dihedral angles: A137914 (tetrahedron), A019669 (cube), A236367 (icosahedron), A137218 (dodecahedron). - _Stanislav Sykora_, Jan 23 2014
%K A156546 nonn,cons
%O A156546 1,2
%A A156546 _Clark Kimberling_, Feb 09 2009