the discrete Gauss-Bonnet theorem
There is a single rule binding the bumps and saddles of a closed surface to a counting number that never changes no matter how you bend it. The discrete Gauss-Bonnet theorem says: add up the angle defect at every vertex of a closed triangulated surface and you always get the same answer, 2*pi times the Euler characteristic — no matter how coarse or fine the mesh, no matter how you dent it. The total curvature is a topological constant. Descartes essentially knew this for polyhedra centuries before Gauss-Bonnet was stated smoothly.
Stated cleanly, for a closed (boundary-free) triangulated surface K, the sum over all vertices v of the angle defect d(v) = 2*pi - (sum of incident corner angles) equals 2*pi * chi(K), where chi(K) = V - E + F is the Euler characteristic. You can prove it by a one-line accounting trick: the total of all corner angles across all triangles is, on one hand, the sum over triangles (each triangle's three angles add to pi, giving pi*F), and on the other hand the sum over vertices of their incident angles, which is 2*pi*V minus the total defect. Setting these equal and using 3F = 2E (each triangle has three edges, each interior edge shared by two triangles) recovers 2*pi(V - E + F). The theorem is the exact discrete shadow of the smooth Gauss-Bonnet theorem, integral of K over M plus boundary term equals 2*pi*chi(M), with angle defect playing the role of integrated Gaussian curvature.
Its importance is conceptual and practical at once. Conceptually it is the cleanest example of curvature being topologically constrained: you may push curvature around — flatten one vertex by sharpening another — but the total is fixed by the genus. Practically it certifies discrete curvature computations: if your code sums vertex defects and does not return 2*pi*chi, you have a bug or a non-closed mesh. A boundary version adds the discrete geodesic curvature of the boundary (turning angles) to the vertex defects; the honest caveat is that this exactness is a property of the combinatorics and edge lengths, so it holds for any triangulation, but it tells you nothing extra beyond the genus — two very different geometries with the same chi have the same total defect.
Take any triangulated torus (genus 1). Then chi = 0, so the discrete Gauss-Bonnet theorem forces the total angle defect to be exactly 0: every positively curved vertex (defect > 0) is balanced by saddle-like vertices (defect < 0), summing to zero regardless of how the donut is meshed or shaped.
On a torus the bumps and saddles must cancel exactly, because chi = 0 nails the total defect to zero.
The identity is exact for every triangulation, but it is purely topological: equal Euler characteristic forces equal total defect, so the theorem never distinguishes geometry. It is a checksum on curvature, not a measurement of shape.