As far as I know Euler's theorem is not restricted to platonic solids, is it? And what does "hollowed out space" mean? --Seb

It applies to any convex solid with planar faces, and no holes (ie, not toroidal etc.) not just Platonic solids. I don't know whether it applies to concave shapes, though - Malcolm Farmer

It does apply to a large number of concave shapes. The convex assumption is too conservative. --Seb

I applies to all polyhedra that, when you "blow them up" and "smooth them out", look like a sphere. So the polyhedron can have "dents", because they will go away when "blowing up", but it cannot have holes like a torus for example. Also, like a sphere, the polyhedron has to be connected. --AxelBoldt