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:
- Using the center of one pentagon as north pole, the longitude and latitude for the centers of the circles
- the longitude and latitude of the centers of the pentagons
- 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.