structural:summarization, structural:scholar_edition, structural:expansion, information:classification
Polyhedra

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

where V, E, and F are respectively the numbers of v ertices (corners), e dges and f aces in the given polyhedron. Any convex polyhedron 's surface has Euler characteristic

This equation, stated by Euler in 1758, is known as Euler's polyhedron formula. It corresponds to the Euler characteristic of the sphere (i.e. χ = 2 {\displaystyle \ \chi =2\ }), and applies identically to spherical polyhedra. An illustration of the formula on all Platonic 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 density D, vertex figure density d v , {\displaystyle \ d_{v}\,} and face density d f : {\displaystyle \ d_{f}\ :}

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

Projective polyhedra all have Euler characteristic 1, like the real projective plane, while the surfaces of toroidal polyhedra all have Euler characteristic 0, like the torus.