due to Euler
(in fact, a special case of the Euler characteristic
). Take any 3 dimensional polytope
(i.e. a solid
with planar face
s) with no hole
s in it. Count its 0-, 1-, and 2-dimensional features:
For instance, consider a regular dodecahedron. It has 12 faces (F=12), each a pentagon, for a total of 30 edges (recall that each edge is shared by 2 faces, so E=30). 3 faces meet at each vertex, so there are 20 vertices (V=20). Amazingly, 12-30+20=2, as promised.
See also Euler formula.