I was vaguely thinking about the problem of tessellating the sphere, noting that the conventional lines of latitude and longitude (which seem to mostly partition the sphere into spherical quadrilaterals except around the poles) doesn't count, as they come in different sizes.
A brief web search brought me to a page which suggested taking the five regular convex platonic solids, surrounding them with a sphere, and then projecting their lines and vertices onto the enclosing sphere. There is a fancy Java applet animation here where you can choose between the five solids.
I don't know if there is a way to tessellate the sphere with an arbitrary number of spherical polygons though.
