Portions of this proof are from Bishop and Goldberg.
Define a height function on the surface of the polyhedron as follows: Choose arbitrary heights for each vertex. In each edge, choose a height for one interior point greater than that of the two endpoints, and interpolate the remainder of the edge linearly between the chosen point and the endpoints. In each face, choose a height for one interior point, greater than all heights on its boundary; interpolate the heights in the rest of the face linearly on line segments from the chosen point to the boundary. The result is a continuous function with \(V+F+E\) critical points: \(V\) local minima at the vertices, \(E\) saddles at the chosen points of edges, and \(F\) local maxima at chosen points of faces.
Now view the surface as an initially bone dry earth on which there is about to fall a deluge which ultimately covers the highest peak. We count the number of lakes and connected land masses formed and destroyed in this rainstorm to obtain the result.
For each local minimum there will be one lake formed. For each saddle there will either be two lakes joined or a single lake doubling back on itself and disconnecting one land mass from another (let \(J\) and \(D\) denote the number of times these events happen respectively). For each peak a land mass will be eliminated. Initially there is one land mass, and in the final sitation there is one global lake. Thus we have \(1+D-F=0\) and \(0+V-J=1\). Combining these two equations with the fact that \(D+J=E\) yields the result.
One can either view the rainfall as (unnaturally) causing the global water levels to always be at the same height, so that two lakes reach a saddle at the same time; or one can take a more realistic viewpoint and say that the rainfall may vary arbitrarily over the globe, but when one lake reaches a saddle the water will spill over it (and the lake will not rise) until the lake on the other side of the saddle reaches the same height.
This proof is close to self-dual, the biggest asymmetry being in the definition of the height function. As usual, the Jordan curve theorem is involved, in the fact that a lake doubling back on itself creates an island. The principle of classifying the singular points (peaks, saddles, and valleys) for a height function defined on a surface is the main idea behind Morse theory, but this proof dates back much earlier than Morse, to an 1863 publication of Möbius.
Proofs of Euler's Formula.
From the Geometry Junkyard, computational and recreational geometry pointers.
David Eppstein, Theory Group, ICS, UC Irvine.