A
formula due to
Euler (in fact, a special case of the
Euler characteristic in
R3). Take any
3 dimensional polytope (i.e. a
solid with
planar faces) with no
holes in it. Count its 0-, 1-, and 2-dimensional features:
Then
V-
E+
F=
2. Always.
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.