Let a closed surface have genus 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 Betti number of the space.
See also Chromatic Number ,
Euler Number ,
Map Coloring ,
Poincaré Formula ,
Polyhedral Formula Explore with Wolfram|Alpha References Alexandroff, P. S. Combinatorial Topology. Armstrong, M. A. "Euler Characteristics." §7.3 in Basic Topology, rev. ed. Regular Polytopes, 3rd ed. Dodson, C. T. J. and Parker, P. E. A User's Guide to Algebraic Topology. Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Henle, M. A Combinatorial Introduction to Topology. Referenced on Wolfram|Alpha Euler Characteristic Cite this as: Weisstein, Eric W. "Euler Characteristic." From MathWorld https://mathworld.wolfram.com/EulerCharacteristic.html
Subject classifications 
. Then the polyhedral formula generalizes to the Poincaré formula 
. 
, 
is the 
th Betti number of the space. 