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Euler Characteristic

Let a closed surface have Genus $g$. Then the Polyhedral Formula becomes the Poincaré Formula

\chi\equiv V-E+F=2-2g,
\end{displaymath} (1)

where $\chi$ is the Euler characteristic, sometimes also known as the Euler-Poincaré Characteristic. In terms of the Integral Curvature of the surface $K$,
\int\!\!\!\int K\,da=2\pi\chi.
\end{displaymath} (2)

The Euler characteristic is sometimes also called the Euler Number. It can also be expressed as
\end{displaymath} (3)

where $p_i$ is the $i$th Betti Number of the space.

See also Chromatic Number, Map Coloring

© 1996-9 Eric W. Weisstein