A156547 - OEIS

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

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OFFSET

1,1

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.

LINKS

Table of n, a(n) for n=1..105.

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)).