A Proof of Euler's Formula

steps

V - E + F = 15 - 20 + 7 = 2

Euler's formula states that for a map on the sphere,

V-E+F=2

, where

V

is the number of vertices,

F

is the number of faces, and

E

is the number of edges. This Demonstration shows a map in the plane (so the exterior face counts as a face). The formula is proved by deleting edges lying in a cycle (which causes

E

and

F

to each decrease by one) until there are no cycles left. Then one has a tree, and one can delete vertices of degree one and the edges connected to them until only a point is left. Each such move decreases

V

and

E

by one. So all the moves leave

V-E+F

unchanged, but at the end

V

and

F

are each 1 and

E

is 0, so

V-E+F

must have been 2 at the start.