Euler Characteristic:

The Euler characteristic of a closed surface is a topological invariant that can be computed in several ways. Two important ones are by counting critical points (the Euler characteristic is the number of maxima and minima minus the number of saddles) and by counting vertices, edges and faces of a polyhedral surface (the Euler characteristic is the number of vertices and faces minus the number of edges).

The Euler characteristic is a fundamental value: this number uniquely classifies closed surfaces up to orientability. That is, given the Euler characteristic and orientability of a surface, the topological type of the surface is determined. This makes the Euler characteristic a powerful computational tool.

11/1/94 dpvc@geom.umn.edu -- The Geometry Center