Euler’s Polyhedron Formula

Abstract

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 Poincar´e’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 −N(−1),+N(0),−N(1),...,〖(-1)〗^(dim(p)) ∗ N(dim(p)), 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 “po lyhedron” 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 Poincar´e’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.