The Polyhedral Formula
Leonhard Euler: Life, Work, and Legacy
On November 14, 1750 Leonhard Euler sent a letter from Berlin to his friend Christian Goldbach in St. Petersburg announcing his discovery of a simple relationship between the features on a polyhedron. This observation, now known as Euler's polyhedral formula, is one of the most beloved theorems in mathematics. (A 1990 survey of mathematicians found the polyhedral formula to be the second most beautiful theorem in all of mathematics.) Euler stated the theorem as follows.
THEOREM: In every solid enclosed by plane faces, the number of faces along with the number of solid angles exceeds the number of edges by two.
This relationship is typically expressed as F - E + V = 2 where F, E, and V denote the number of faces, edges, and vertices of a polyhedron.
Richeson, David. "The Polyhedral Formula." In Leonhard Euler: Life, Work, and Legacy, edited by Robert E. Bradley and C. Edward Sandifer, 421-439. Amsterdam: Elsevier, 2007.