The Euler characteristic of a
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
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.
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The Geometry Center