# Deforming a truncated icosahedron into its circumscribing sphere

Imagine that I have a truncated icosahedron consisted of 60 vertices, each of degree $deg(v) = 3$, and fixed edge length $L$. I'd like to assign some constant curvature or bending angle $\theta$ to each edge s.t. I can deform the icosahedron into its circumscribing sphere.

As a function of the edge length $L$, what value of $\theta$ allows me to properly perform this deformation?

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I don't understand how you're using the "bending angle" to deform the icosahedron. Are you looking for the angle $\theta$ that each edge makes with the circumscribing sphere? –  anon Jul 17 '11 at 23:49
@anon, sorry for the confusion! No I'm looking for the bending angle that places each edge on the surface of the sphere. –  R.H. Jul 18 '11 at 3:04
Oh, you mean the angle formed by the arc which results from a radial projection of an edge onto the circumscribing sphere. –  anon Jul 19 '11 at 12:15

Fix $L=1$. Then the radius of the circumscribed sphere is $r=\frac{1}{4} \sqrt{58+18 \sqrt{5}} \approx 2.478$. Now look at the isosceles triangle formed by the center of the sphere and one edge. It has sides of length $r$ and base length 1. So the angles at either end of the base are $\cos^{-1} (1/(2r)) \approx 78.3593^\circ$. The angle between the tangent to the sphere at one endpoint and the edge is then about $11.6407^\circ$.