Angles between two vertices on a dodecahedron Say $20$ points are placed across a spherical planet, and they are all spaced evenly, forming the vertices of a dodecahedron. I would like to calculate the distances between the points, but that requires me to find out the angles between the vertices.
From the origin of the dodecahedron, how would I find the angle between two adjacent vertices on the same face, and the angle between two vertices on the same face but not connected by an edge?

There are 3 angles to consider in a polygon involving the origin point. the circumsphere radius (vertices) what you are seeking, the midsphere (center of edges, aka. 1/2 dihedral angle), and the insphere (right angle perpendicular to the face).
Formulas for finding these three radii are:
C = a/4 * √3 * (1 + √5)
M = a/4 * (3 + √5)
I = a/2 * √([25 + {11*√5}]/10)
Also useful to know the radii formulas for the pentagon.
c = a/10 * √(50 + [10*√5]) 
i = a/10 * √(25 + [10*√5])
Then you just solve for the right triangle OIC, where you have the three sides and the right angle, height=I, base=c, and hypotenuse=C, just have to find the other two angles.
