Polyhedral Formula

A formula relating the number of Vertices, Faces, and Edges of a Polyhedron (or Polygon). It was discovered independently by Euler and Descartes, so it is also known as the Descartes-Euler Polyhedral Formula. The polyhedron need not be Convex, but the Formula does not hold for Stellated Polyhedra.

$$V+F-E=2,$$ (1)

where $V=N_0$ is the number of Vertices, $E=N_1$ is the number of Edges, and $F=N_2$ is the number of Faces. For a proof, see Courant and Robbins (1978, pp. 239-240). The Formula can be generalized to $n$-D Polytopes.

$\quad \Pi_1: N_0=2$	(2)
$\quad \Pi_2: N_0-N_1=0$	(3)
$\quad \Pi_3: N_0-N_1+N_2=2$	(4)
$\quad \Pi_4: N_0-N_1+N_2-N_3=0$	(5)
$\quad \Pi_n: N_0-N_1+N_2-\ldots+(-1)^{n-1}N_{n-1}= 1-(-1)^n.$	(6)

For a proof of this, see Coxeter (1973, pp. 166-171).

See also Dehn Invariant, Descartes Total Angular Defect

References

Beyer, W. H. (Ed.) CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 128, 1987.

Courant, R. and Robbins, H. What is Mathematics?: An Elementary Approach to Ideas and Methods. Oxford, England: Oxford University Press, 1978.

Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York: Dover, 1973.