A

formula due to

Euler (in fact, a special case of the

Euler characteristic in

**R**^{3}). 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.