Let a closed surface have genus . Then the polyhedral formula generalizes to the Poincaré formula

(1)

where

(2)

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

The only compact closed surfaces with Euler characteristic 0 are the Klein bottle and torus (Dodson and Parker 1997, p. 125). The following table gives the Euler characteristics for some common surfaces (Henle 1994, pp. 167 and 295; Alexandroff 1998, p. 99).

surface
cylinder	0
double torus
Klein bottle	0
Möbius strip	0
projective plane	1
sphere	2
torus	0

In terms of the integral curvature of the surface ,

(3)

The Euler characteristic is sometimes also called the Euler number. It can also be expressed as

(4)

where is the th Betti number of the space.

Alexandroff, P. S. Combinatorial Topology. New York: Dover, 1998.

Armstrong, M. A. "Euler Characteristics." §7.3 in Basic Topology, rev. ed. New York: Springer-Verlag, pp. 158-161, 1997 Coxeter, H. S. M. "Poincaré's Proof of Euler's Formula." Ch. 9 in Regular Polytopes, 3rd ed. New York: Dover, pp. 165-172, 1973.

Dodson, C. T. J. and Parker, P. E. A User's Guide to Algebraic Topology. Dordrecht, Netherlands: Kluwer, 1997.

Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, p. 635, 1997.

Henle, M. A Combinatorial Introduction to Topology. New York: Dover, p. 167, 1994.

CITE THIS AS:

Weisstein, Eric W. "Euler Characteristic." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/EulerCharacteristic.html