Given are circles of a certain radius $r_c$. These are arranged on a sphere with the radius $r_s$ in a pattern of hexagons and pentagons similar to a soccer ball (truncated icosahedron). Five circles are fitted into a pentagon and seven into a hexagon.

In the center of the sphere sits an icosahedron. A line is drawn from each of its corners to the center of each pentagon.

What is needed:

  1. Using the center of one pentagon as north pole, the longitude and latitude for the centers of the circles
  2. the longitude and latitude of the centers of the pentagons
  3. The radius of the sphere.

To illustrate the matter, pictures of the object in question can be found here: http://i.imgur.com/pU4pmOp.jpg


To build the real life object I calculated the length of the sides of a hexagon using $r_c$. Using the side length I could calculate the circumscribing sphere of a truncated icosahedron. But since the base for this calculation puts the circles on a plane and not the sphere itself, it is neither accurate nor does it give me the coordinates.


I am building a ball of speakers. I have already done that IRL but now I want to construct a CAD model. Where real life wire can be bent to fit, CAD is merciless. So I need to get to the bottom of this mathematically.


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