Let a closed surface have Genus
. Then the Polyhedral Formula becomes the
Poincaré Formula
![\begin{displaymath}
\chi\equiv V-E+F=2-2g,
\end{displaymath}](e_2048.gif) |
(1) |
where
is the Euler characteristic, sometimes also known as the Euler-Poincaré
Characteristic. In terms of the Integral Curvature of the surface
,
![\begin{displaymath}
\int\!\!\!\int K\,da=2\pi\chi.
\end{displaymath}](e_2050.gif) |
(2) |
The Euler characteristic is sometimes also called the Euler Number. It can also be expressed as
![\begin{displaymath}
\chi=p_0-p_1+p_2,
\end{displaymath}](e_2051.gif) |
(3) |
where
is the
th Betti Number of the space.
See also Chromatic Number, Map Coloring
© 1996-9 Eric W. Weisstein
1999-05-25