Lecture 8: Mesh Representations and Geometry Processing (43)
andrewhuang56
More information regarding "Euler's Convex Polyhedron Formula," also known as the Euler characteristic, can be found here on Wikipedia. The proof that I know, which I believe is also the one on Wikipedia, involves induction on the number of edges.
Now, note that this is not true of shapes that are not topologically equivalent to spheres (i.e. have holes, like toruses). For that, the formula is 2-2n, where n is the number of holes. So, a (one hole) torus can have all degree 6 vertices.
More information regarding "Euler's Convex Polyhedron Formula," also known as the Euler characteristic, can be found here on Wikipedia. The proof that I know, which I believe is also the one on Wikipedia, involves induction on the number of edges. Now, note that this is not true of shapes that are not topologically equivalent to spheres (i.e. have holes, like toruses). For that, the formula is 2-2n, where n is the number of holes. So, a (one hole) torus can have all degree 6 vertices.