Editing Euler characteristic (section)

===Proof of Euler's formula===
[[Image:V-E+F=2 Proof Illustration.svg|frame|right|First steps of the proof in the case of a cube]]

There are many proofs of Euler's formula. One was given by [[Augustin Louis Cauchy|Cauchy]] in 1811, as follows. It applies to any convex polyhedron, and more generally to any polyhedron whose boundary is topologically equivalent to a sphere and whose faces are topologically equivalent to disks.

Remove one face of the polyhedral surface. By pulling the edges of the missing face away from each other, deform all the rest into a planar graph of points and curves, in such a way that the perimeter of the missing face is placed externally, surrounding the graph obtained, as illustrated by the first of the three graphs for the special case of the cube. (The assumption that the polyhedral surface is homeomorphic to the sphere at the beginning is what makes this possible.) After this deformation, the regular faces are generally not regular anymore. The number of vertices and edges has remained the same, but the number of faces has been reduced by&nbsp;1. Therefore, proving Euler's formula for the polyhedron reduces to proving <math>\ V - E + F = 1\ </math> for this deformed, planar object.

If there is a face with more than three sides, draw a diagonal—that is, a curve through the face connecting two vertices that are not yet connected. Each new diagonal adds one edge and one face and does not change the number of vertices, so it does not change the quantity <math>\ V - E + F ~.</math> (The assumption that all faces are disks is needed here, to show via the [[Jordan curve theorem]] that this operation increases the number of faces by one.) Continue adding edges in this manner until all of the faces are triangular.

Apply repeatedly either of the following two transformations, maintaining the invariant that the exterior boundary is always a [[simple cycle]]:
#Remove a triangle with only one edge adjacent to the exterior, as illustrated by the second graph. This decreases the number of edges and faces by one each and does not change the number of vertices, so it preserves <math>\ V - E + F ~.</math>
#Remove a triangle with two edges shared by the exterior of the network, as illustrated by the third graph. Each triangle removal removes a vertex, two edges and one face, so it preserves <math>\ V - E + F ~.</math>
These transformations eventually reduce the planar graph to a single triangle. (Without the simple-cycle invariant, removing a triangle might disconnect the remaining triangles, invalidating the rest of the argument. A valid removal order is an elementary example of a [[shelling (topology)|shelling]].)

At this point the lone triangle has <math>\ V = 3\ ,</math><math>\ E = 3\ ,</math> and <math>\ F = 1\ ,</math> so that <math>\ V - E + F = 1 ~.</math> Since each of the two above transformation steps preserved this quantity, we have shown <math>\ V - E + F = 1\ </math> for the deformed, planar object thus demonstrating <math>\ V - E + F = 2\ </math> for the polyhedron. This proves the theorem.

For additional proofs, see [[David Eppstein|Eppstein]] (2013).<ref name=Eppstein2013>{{cite web |last=Eppstein |first=David |year=2013 |title=Twenty-one proofs of Euler's formula: {{nobr|{{math| V − E + F {{=}} 2 }} }} |type=acad. pers. wbs. |url=http://www.ics.uci.edu/~eppstein/junkyard/euler/ |access-date=27 May 2022 |via=[[University of California, Irvine|UC Irvine]] }}</ref> Multiple proofs, including their flaws and limitations, are used as examples in ''[[Proofs and Refutations]]'' by [[Imre Lakatos|Lakatos]] (1976).<ref>{{cite book |author=Lakatos, I. |author-link=Imre Lakatos |year=1976 |title=Proofs and Refutations |title-link=Proofs and Refutations |publisher=Cambridge Technology Press}}</ref>