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
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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.
