A Congruence Problem for Polyhedra

We will make this more specific by specifying what sorts of measurements will be allowed. For example, in much of the paper, allowed measurements will include distances between pairs of vertices, angles between edges, angles between two intersecting face diagonals (possibly on different faces with a common vertex) or between a face diagonal and an edge, and dihedral angles (that is, angles between two adjoining faces). One motivation for these choices is given below. Sometimes we are more restrictive, for example, allowing only distance measurements. In two dimensions this was a fundamental question answered by Euclidean geometers, as (we hope) every student who takes geometry in high school learns. If the lengths of the corresponding sides of two triangles are equal, then the triangles are congruent. The SAS, ASA, and AAS theorems are equally well known. The extension to other shapes is not often discussed, but we will have some remarks about the planar problem as well. It is surprising to us that beyond the famous theorem of Cauchy discussed below, we have been unable to find much discussion of the problems we consider in the literature, though we think it is almost certain that they have been studied in the past. We would be appreciative if any reader can point us to relevant results. Our approach will usually be to look at the problem locally. If the two polyhedra are almost congruent, and agree in a certain set of measurements, are they congruent? At first glance this looks like a basic question in what is known as rigidity theory, but a little thought shows that it is different. In rigidity theory, attention is paid to relative positions of vertices, viewing these as connected by inextensible rods which are hinged at their ends and so can rotate relative to each other, subject to constraints imposed by the overall structure of rods. In our problem there is the additional constraint that in any movement of the vertices, the combinatorial structure of the polyhedron cannot change. In particular, any vertices that were coplanar before the movement must be coplanar after the movement. This feature seems to us to produce an interesting area of study. Our original motivation for considering this problem came from a very practical question encountered by one of us (SPH). If one attempts to make solid wooden models of interesting polyhedra, using standard woodworking equipment, it is natural to want to check how accurate these models are.1 As a mathematician one may be attracted first to the Platonic solids, and of these, the simplest to make appears to be the cube. (The regular tetrahedron looks harder, because non-right angles seem harder to cut accurately.)