Euler's polyhedron formula

Solved by Leonhard Euler, with later proofs by Cauchy and others.

Euler stated the formula in 1758, and it is easy to test on examples, for instance a cube has $8-12+6=2$. Many proofs are now known. A classical one, going back to Cauchy, removes one face and flattens the rest into a planar graph, then deletes edges one at a time while checking that $V-E+F$ does not change. The quantity $V-E+F$ is the first example of a topological invariant, now called the Euler characteristic. It equals 2 for every surface that can be deformed into a sphere, 0 for a torus and $2-2g$ for a closed orientable surface with $g$ holes.

References

• Euler characteristic (Wikipedia)