Convex Polytopes

The study of convex polytopes in Euclidean space of two and three dimensions is one of the oldest branches of mathematics. Yet many of the more interesting properties of polytopes have been discovered comparatively recently, and are still unknown to the majority of mathematicians. In this paper we shall survey the subject, mentioning some of the most recent results, and stating the more important unsolved problems. In order to make the exposition as self-contained as possible, in §2.1 and §3.1 we give a number of definitions with which the reader may not be familiar. We have separated our account of the combinatorial properties of polytopes in §2, from those of a metrical character in §3. As will be seen, these two aspects of the subject overlap, and distinguishing between them is, to a large extent, a matter of expository convenience only. It is beyond the scope of the present survey to indicate proofs of the results mentioned. For these the reader is referred either to the original papers, or, for the earlier results, to the standard textbooks. In the notes references are given to recent publications, and in these, precise references to the older literature may be found. It has, of course, been necessary to make a small selection from the vast amount of published material. To some extent the selection has been made according to the authors' personal interests, but we hope that it nevertheless represents a reasonably balanced account of our subject. Before the beginning of this century, three events can be picked out as being of the utmost importance for the theory of convex polytopes. The first was the publication of Euclid's Elements which, as Sir D'Arcy Thompson once remarked, was intended as a treatise on the five regular (Platonic) 3-polytopes, and not as an introduction to elementary geometry. The second was the discovery in the eighteenth century of the celebrated Euler's Theorem (see §2.2) connecting the numbers of vertices, edges and polygonal faces of a convex polytope in E. Not only is this a result of great generality, but it initiated the combinatorial theory of polytopes. The third event occurred about a century later with the discovery of polytopes in d ^ 4 dimensions. This has been attributed to the Swiss mathematician Ludwig Schlafli; it happened at a time when very few mathematicians (Cayley, Grassmann, Mobius) realised that geometry in more than three dimensions was possible. During the latter hah of the nineteenth century a large'amount of work concerning polytopes was done, mostly extending the earlier metrical work to d ^ 4 dimensions. The symmetry groups of polytopes were extensively studied (see §3.4) and the