Euler Characteristic

The Euler characteristic was classically defined for the surfaces of polyhedra, according to the formula

where V, E, and F are respectively the numbers of vertices (corners), edges and faces in the given polyhedron. Any convex polyhedron's surface has Euler characteristic

This result is known as Euler's polyhedron formula or theorem. It corresponds to the Euler characteristic of the sphere (i.e. χ = 2), and applies identically to spherical polyhedra. An illustration of the formula on some polyhedra is given below.

The surfaces of nonconvex polyhedra can have various Euler characteristics;

For regular polyhedra, Arthur Cayley derived a modified form of Euler's formula using the densities of the polyhedron D, vertex figures and faces :

This version holds both for convex polyhedra (where the densities are all 1), and the non-convex Kepler–Poinsot polyhedrons:

Projective polyhedra all have Euler characteristic 1, corresponding to the real projective plane, while toroidal polyhedra all have Euler characteristic 0, corresponding to the torus.