Can anyone offer insight on an abstract math problem?

Hello members of DIYAudio,

I am a bit stumped on a mathematical problem. Recently, I've become interested in polyhedrons. However, I am having a bit of trouble with their analysis.

Consider a given dodecahedron. If we consider the solid to be of uniform density, the centroid will occur at (0,0,0) and the distance from the centroid to all of the vertices will be constant. However, the distance from the centroid to other points on the surface will vary. Since the dodecahedron is symmetrical, we can restrict the problem to a single pentagon on the surface.

I would like to plot a 2 dimensional function (y=f(x)), where "x" represents radius and "y" represents frequency. For the dodecahedron, the domain of "x" (ie radius) will be continuous from a vertex to the center of the pentagon. The domain of "y" will be continuous from 0-->1 (ie it will be normalized).

I would like to extend this analysis to assymetrical polyhedron (ex. the diminished rhombicosidodecahedron. In that particular case, the centroid will be displaced and the individual polygons will not be seperable.

Would the solution simply be analogous to a fourier transform over the surface?

Could anyone offer insight on this?