A156547 - OEIS

%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