Proof of Euler formula base on the idea of W. P. Thurston.
Here, red dot represent positive charge and blue dot represent negative charge.
Add all up will give us the number of vertices + the number of faces - the number of edges, this number is called Euler characteristic.
For convex polyhedron with convex faces, we see its Euler characteristic equals 2.
For detail, please visit:
http://www.ics.uci.edu/~eppstein/junkyard/euler/charges.html
