A156547 Decimal expansion of the central angle of a regular dodecahedron.

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, 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, 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

Offset: 1

Author

Clark Kimberling, Feb 09 2009

Keywords

• nonn
• cons

Comments

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).

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

- tetrahedron: A156546,

- cube: A137914,

- octahedron: A019669,

- icosahedron: A105199.

Examples

			arccos((1/3)*sqrt(5))=0.729727656226966..., or, in degrees,
41.810314895778598065857916730578259531014119535901347753...
		

Crossrefs

Cf. A001622.

Cf. A019669, A105199, A137914, A156546.

Programs

Maple

evalf(arcsin(2/3)); #  Robert FERREOL, Sep 14 2019

Mathematica

RealDigits[ArcCos[Sqrt[5]/3],10,120][[1]] (* Harvey P. Dale, Feb 23 2015 *)

PARI

asin(2/3) \\ Charles R Greathouse IV, May 28 2013

Formula

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:

- (d,d,d) where d is 1 or -1 (that's 8 vertices);

- (0, d*(t-1),d*t), where d is 1 or -1 and d = golden ratio = (1+sqrt(5))/2;

- (d*(t-1), d*t, 0); and ((d*t,0,d*(t-1)).

An example of a neighboring pair is (1,1,1) and (0,t,t-1).

Apply the usual formula for the cosine of the angle between two vectors.

Equals 2 * arccot(phi^2), where phi is the golden ratio (A001622). - Amiram Eldar, Jul 06 2023