FormulaProved

Euler's polyhedron formula

Statement

For any convex polyhedron with $V$ vertices, $E$ edges, and $F$ faces, $V - E + F = 2$ .

Why is it true?

Flatten a polyhedron's surface onto a plane (remove one face and stretch the rest flat) to get a connected planar graph; V - E + F stays the same no matter how you build up or simplify that graph, because adding a vertex on an edge, or an edge splitting a face, changes two of the three counts by +1 and -1 at once.

Proof sketch

Project the polyhedron's edges onto a sphere around an interior point, then puncture the sphere through the interior of one face and flatten it into a planar graph. Triangulate every non-triangular face by adding diagonals (each diagonal adds one edge and one face, leaving V - E + F unchanged), then repeatedly remove an outer edge shared by two triangles - this removes either one edge and one face, or one vertex, two edges and one face, again preserving V - E + F. The process ends with a single triangle, for which V - E + F = 3 - 3 + 1 = 1; adding back the removed face gives V - E + F = 2.

Stated by

stated 1750Leonhard Euler

Proved by

proved 1813Leonhard Euler proved 1813Augustin-Louis Cauchy

Topics that use this theorem

Euler characteristic

Related theorems

Hairy ball theorem Classification of closed surfaces Gauss–Bonnet theorem

Step-by-step proofs

Cauchy's projection–triangulation proof of V − E + F = 2 (1811)

Augustin-Louis Cauchy, 1811

References

David S. Richeson (2008). Euler's Gem: The Polyhedron Formula and the Birth of Topology
Augustin-Louis Cauchy (1813). Recherches sur les polyèdres — première partie