Euler's Polyhedron Formula (1758): For any convex polyhedron, V − E + F = 2.
This is not a coincidence — it reflects the topology of the sphere (genus 0).
General formula: χ = 2 − 2g, where g is the number of handles (genus).
Sphere: χ=2 (g=0) | Torus: χ=0 (g=1) | Double torus: χ=−2 (g=2)
Gauss-Bonnet theorem: The integral of Gaussian curvature K over a closed surface equals 2πχ:
∬ K dA = 2πχ = 2π(2−2g)
Positive curvature (sphere): K>0 everywhere, total = 4π.
Torus: positive curvature on outer equator, negative on inner — exactly cancel to 0.
This connects local geometry to global topology — a profound bridge in differential geometry.