Nineteen Proofs of Euler's Formula: V-E+F=2

Many theorems in mathematics are important enough that they have been proved repeatedly in surprisingly many different ways. Examples of this include the existence of infinitely many prime numbers, the evaluation of zeta(2), the fundamental theorem of algebra (polynomials have roots), quadratic reciprocity (a formula for testing whether an arithmetic progression contains a square) and the Pythagorean theorem (which according to Wells has at least 367 proofs). This also sometimes happens for unimportant theorems, such as the fact that in any rectangle dissected into smaller rectangles, if each smaller rectangle has integer width or height, so does the large one.

This page lists proofs of the Euler formula: for any convex polyhedron, the number of vertices and faces together is exactly two more than the number of edges. Symbolically V-E+F=2. For instance, a tetrahedron has four vertices, four faces, and six edges; 4-6+4=2.

According to Malkevitch, this formula was discovered in around 1750 by Euler, and first proven by Legendre in 1794. Earlier, Descartes (around 1639) discovered a related polyhedral invariant (the total angular defect) but apparently did not notice the Euler formula itself. Hilton and Pederson provide more references as well as entertaining speculation on Euler's discovery of the formula. Confusingly, other equations such as e^i pi = -1 and a^phi(n) = 1 (mod n) also go by the name of "Euler's formula"; Euler was a busy man.

The polyhedron formula, of course, can be generalized in many important ways, some using methods described below. One important generalization is to planar graphs. To form a planar graph from a polyhedron, place a light source near one face of the polyhedron, and a plane on the other side.

The shadows of the polyhedron edges form a planar graph, embedded in such a way that the edges are straight line segments. The faces of the polyhedron correspond to convex polygons that are faces of the embedding. The face nearest the light source corresponds to the outside face of the embedding, which is also convex. Conversely, any planar graph with certain connectivity properties comes from a polyhedron in this way.

Some of the proofs below use only the topology of the planar graph, some use the geometry of its embedding, and some use the three-dimensional geometry of the original polyhedron. Graphs in these proofs will not necessarily be simple: edges may connect a vertex to itself, and two vertices may be connected by multiple edges. Several of the proofs rely on the Jordan curve theorem, which itself has multiple proofs; however these are not generally based on Euler's formula so one can use Jordan curves without fear of circular reasoning.

Please send email if you know of a proof not listed here. I would especially appreciate proofs involving cohomology theory, toric varieties, or other higher mathematics. (Helena Verrill has shown that Euler's formula is equivalent to the fact that every toric variety over GF[p] has a number of points equal to 1 (mod p) but is still missing a nice non-combinatorial proof of the latter fact.)

I imagine it would be possible to construct inductions based on the representation of convex polyhedra as intersections of halfspaces or convex hulls of points, but the need to handle inputs in non-general position would make the resulting proofs quite messy.

There also seems to be a potential connection to binomials: if one defines a polynomial p(t) = 1+Vt+Et²+Ft³+t⁴, the Euler formula can be interpreted as saying that p(t) is divisible by 1+t. But for simplices of any dimension, p(t)=(1+t)^d+1 by the binomial formula. Perhaps there is a proof of Euler's formula that uses these polynomials directly rather than merely translating one of the inductions into polynomial form. Jim Propp asks similar questions for infinite-dimensional polytopes, interpreting p(t) as a power series (see also his recent expansion of these ideas).

From the Geometry Junkyard, computational and recreational geometry pointers.
David Eppstein, Theory Group, ICS, UC Irvine.

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