Euler's V−E+F equals two for every convex polyhedron
The Euler characteristic is a number describing a shape's structure that stays the same however the shape is bent. For a polyhedron's surface it is vertices minus edges plus faces, and every convex polyhedron gives 2, the same value as a sphere—a rule Euler stated in 1758.
Francesco Maurolico had written the relation down for the Platonic solids in an unpublished manuscript of 1537. Leonhard Euler extended it to convex polyhedra in general but did not rigorously prove it was invariant. The count was first used to prove theorems about polyhedra, including the classification of the Platonic solids, and today it comes out of homology and homological algebra.
Other shapes give other values. Toroidal polyhedra, shaped like a ring, score 0 like the torus, while projective polyhedra score 1 like the real projective plane. Arthur Cayley adjusted the formula with density terms so that it also covers the star-shaped Kepler–Poinsot polyhedra. For a connected graph drawn in the plane, counting the outside region as a face, the answer is again 2, provable by induction starting from a tree; Cauchy proved it in one of his few graph theory papers.
Cauchy also gave a proof of Euler's formula in 1811. Remove one face, stretch the rest flat into a planar graph, split every face into triangles with diagonals, then peel triangles off the outer edge one at a time. Each step leaves vertices minus edges plus faces unchanged until a single triangle remains. Imre Lakatos used the many proofs of the formula, flaws and all, as case studies in his 1976 book Proofs and Refutations.
In modern terms these surfaces are finite cell complexes, and the characteristic generalises to an alternating sum of cell counts in each dimension. For any topological space it can be defined as the alternating sum of Betti numbers, the ranks of its homology groups, provided they are finite and eventually zero.
Source: Euler characteristic
