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).

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.

See also Chromatic Number ,

Euler Number ,

Map Coloring ,

Poincaré
Formula ,

Polyhedral Formula
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References 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. Referenced
on Wolfram|Alpha Euler Characteristic
Cite this as:
Weisstein, Eric W. "Euler Characteristic."
From MathWorld --A Wolfram Web Resource. https://mathworld.wolfram.com/EulerCharacteristic.html

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