Euler's Polyhedron Formula

Euler's polyhedron theorem states for a polyhedron p, that

V - E + F = 2,

where V, E, and F are, respectively, the number of vertices, edges, and faces of p. The formula was first stated in print by Euler in 1758 [11]. The proof given here is based on Poincaré's linear algebraic proof, stated in [17] (with a corrected proof in [18]), as adapted by Imre Lakatos in the latter's Proofs and Refutations [15].

As is well known, Euler's formula is not true for all polyhedra. The condition on polyhedra considered here is that of being a homology sphere, which says that the cycles (chains whose boundary is zero) are exactly the bounding chains (chains that are the boundary of a chain of one higher dimension).

The present proof actually goes beyond the three-dimensional version of the polyhedral formula given by Lakatos; it is dimension-free, in the sense that it gives a formula in which the dimension of the polyhedron is a parameter. The classical Euler relation V - E + F = 2 is corresponds to the case where the dimension of the polyhedron is 3.

The main theorem, expressed in the language of the present article, is

Sum alternating - characteristic - sequence (p) = 0,

where p is a polyhedron. The alternating characteristic sequence of a polyhedron is the sequence

where N(k) is the number of polytopes of p of dimension k. The special case of dim(p) = 3 yields Euler's classical relation. (N(-1) and N(3) will turn out to be equal, by definition, to 1.)

Two other special cases are proved: the first says that a one-dimensional "polyhedron" that is a homology sphere consists of just two vertices (and thus consists of just a single edge); the second special case asserts that a two-dimensional polyhedron that is a homology sphere (a polygon) has as many vertices as edges.

A treatment of the more general version of Euler's relation can be found in [12] and [6]. The former contains a proof of Steinitz's theorem, which shows that the abstract polyhedra treated in Poincaré's proof, which might not appear to be about polyhedra in the usual sense of the word, are in fact embeddable in R³ under certain conditions. It would be valuable to formalize a proof of Steinitz's theorem and relate it to the development contained here.

MML identifier: POLYFORM, version: 7.8.05 4.89.993