χ = 0 − 0 + 0 = 0
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Topological invariant — χ never changes on same surface type!
The Euler characteristic is a topological invariant. For any triangulation of a surface, V − E + F = χ — independent of how you triangulate. Sphere: χ=2, torus: χ=0, double torus: χ=−2.
Discovered by Euler (1752) for polyhedra, generalized by Poincaré. For orientable surfaces: χ=2−2g where g is the genus (number of handles). The Gauss-Bonnet theorem connects χ to Gaussian curvature: ∮K dA = 2πχ.