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%I #20 Jul 06 2023 01:59:48
%S 7,2,9,7,2,7,6,5,6,2,2,6,9,6,6,3,6,3,4,5,4,7,9,6,6,5,9,8,1,3,3,2,0,6,
%T 9,5,3,9,6,5,0,5,9,1,4,0,4,7,7,1,3,6,9,0,7,0,8,9,4,9,4,9,1,4,6,1,8,1,
%U 8,8,9,9,6,6,6,7,6,7,1,3,8,7,9,5,4,8,3,4,0,7,8,1,9,4,7,3,5,0,0,2,0,8,0,9,5
%N Decimal expansion of the central angle of a regular dodecahedron.
%C If A and B are neighboring vertices of a regular dodecahedron having center O, then the central angle AOB is this number; the exact value is arccos((1/3)*sqrt(5)) = arcsin(2/3).
%C The (minimal) central angle of the other four regular polyhedra are as follows:
%C - tetrahedron: A156546,
%C - cube: A137914,
%C - octahedron: A019669,
%C - icosahedron: A105199.
%F The dodecahedron has 12 faces and 20 vertices. To find the central angle, we need any neighboring pair of vertices. Here are all 20 vertices:
%F - (d,d,d) where d is 1 or -1 (that's 8 vertices);
%F - (0, d*(t-1),d*t), where d is 1 or -1 and d = golden ratio = (1+sqrt(5))/2;
%F - (d*(t-1), d*t, 0); and ((d*t,0,d*(t-1)).
%F An example of a neighboring pair is (1,1,1) and (0,t,t-1).
%F Apply the usual formula for the cosine of the angle between two vectors.
%F Equals 2 * arccot(phi^2), where phi is the golden ratio (A001622). - _Amiram Eldar_, Jul 06 2023
%e arccos((1/3)*sqrt(5))=0.729727656226966..., or, in degrees,
%e 41.810314895778598065857916730578259531014119535901347753...
%p evalf(arcsin(2/3)); # _Robert FERREOL_, Sep 14 2019
%t RealDigits[ArcCos[Sqrt[5]/3],10,120][[1]] (* _Harvey P. Dale_, Feb 23 2015 *)
%o (PARI) asin(2/3) \\ _Charles R Greathouse IV_, May 28 2013
%Y Cf. A001622.
%Y Cf. A019669, A105199, A137914, A156546.
%K nonn,cons
%O 1,1
%A _Clark Kimberling_, Feb 09 2009
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