I've always loved the proof of the Gauss-Bonnet theorem, which ties together differential geometry and topology, and basically states that however you distort a

I've always loved the proof of the Gauss-Bonnet theorem, which ties together differential geometry and topology, and basically states that however you distort a compact, boundaryless two-dimensional Riemannian manifold, its total curvature remains the same. The proof (well, the proof I learned -- there are several) was long with many steps, each of which was simple and intuitive. It comes together in a sort of crescendo. Here: http://www.math.dartmouth.edu/~leibon/gbt/sld001.htm http://www.math.dartmouth.edu/~leibon/gbt/sld001.htm