Polyhedral Formula

A formula relating the number of polyhedron vertices , faces , and polyhedron edges of a simply connected (i.e., genus 0) polyhedron (or polygon). It was discovered independently by Euler (1752) and Descartes, so it is also known as the Descartes-Euler polyhedral formula. The formula also holds for some, but not all, non-convex polyhedra.

The polyhedral formula states

(1)

where is the number of polyhedron vertices, is the number of polyhedron edges, and is the number of faces. For a proof, see Courant and Robbins (1978, pp. 239-240).

The formula was generalized to -dimensional polytopes by Schläfli (Coxeter 1973, p. 233),

	(2)
	(3)
	(4)
	(5)
	(6)

and proved by Poincaré (Poincaré 1893; Coxeter 1973, pp. 166-171; Williams 1979, pp. 24-25).

For genus surfaces, the formula can be generalized to the Poincaré formula

(7)

where

(8)

is the Euler characteristic, sometimes also known as the Euler-Poincaré characteristic. The polyhedral formula corresponds to the special case .

There exist polytopes which do not satisfy the polyhedral formula, the most prominent of which are the great dodecahedron and small stellated dodecahedron , which no less than Schläfli himself refused to recognize (Schläfli 1901, p. 134) since for these solids,

(9)

(Coxeter 1973, p. 172).