Uniqueness theorems for polyhedra.

In 1813, Cauchy2 gave the first proof of the theorem that two closed convex polyhedra in three-dimensional space are congruent if their faces are congruent in pairs and are joined to each other in the same order: in effect, the a priori possibility of rotations of the faces about the edges-which are obviously seen to be possible for some easily constructed open polyhedra-cannot occur for closed convex polyhedra. Cauchy's proof was based on two lemmas, one metric, the other topological in character. The proofs of both lemmas were criticized because Cauchy overlooked certain exceptional circumstances in proving preliminary lemmas. More than a hundred years passed before correct, but rather long and complicated, proofs were given by Steinitz (see the books of Steinitz and Rademacher,4 and of Hadamard,3 who credits Lebesgue with the first correct proofs). However, it is rather easy to give proofs that are essentially the same as those of Cauchy but which take care of the two details overlooked by him. The topological Lemma T refers to nets that form the boundaries of a simple covering of a sphere by polygons, or, in other words, the net of edges of a triangulation of the sphere; however, no face polygons are permitted that have only two edges. It is supposed that the edges of such a net are divided into three groups distinguished from one another by giving signs +,-, or 0 to the edges of each of the groups. An index j is assigned to each vertex by making one complete circuit about it and counting the number of sign changes in the edges going out of it from + to -, or to +, while ignoring the signs 0. It is assumed that j = 0 at a vertex only if all edges on that vertex are marked with zeros. It is also assumed that either j = 0, or j . 4 holds at all vertices of the net. The topological Lemma T then states that j = 0 necessarily holds at all vertices, or, in other words, that no edges marked with + or can occur under the circumstances postulated. (This is the author's version of the lemma, rather than Cauchy's; it is less general than Cauchy's, but sufficient for the purposes in view, and easier to prove.) The proof uses Euler's formula for the characteristic of the sphere in terms of a triangulation, together with simple properties of manifolds. A dual Lemma T' is an easy consequence of T, provided that the net is such that at least three edges emanate from all vertices. This lemma refers to an index j defined with respect to circuits around faces, rather than vertices, in the obvious way. It states, in the analogous circumstances, that j = 0 must hold for all faces. The uniqueness theorems for polyhedra to be formulated here are proved by showing that the net of the edges of any such polyhedron can be divided into three groups such that the metric properties at each vertex of the polyhedron give rise to an assignment of its edges to one of these groups. This is done by isolating the vertex with its faces from the rest of the polyhedron and considering deformations of it that are permitted under various circumstances. It is then to be shown that j 2 4 holds at every vertex unless no deformation of the vertex in space occurs, in which case j = 0 holds. It follows then from Lemma T (or sometimes from Lemma