Prologue: Regular polytopes

Current Version January 1, 2005 To motivate the study of finite reflection groups and Coxeter groups more generally, I’ll begin by briefly sketching the classification of regular polytopes. Convex polytopes are fundamental objects in mathematics which can be viewed in a number of equivalent ways: as the convex hull of a finite set of points in R, as the intersection of a finite number of half-spaces whose intersection is compact, or as the image of a high-dimensional simplex under a linear transformation. Within the class of convex polytopes, those which are “completely symmetric” are particularly beguiling; they also have a tendency to play a major role in seemingly disparate areas of mathematics. These highly symmetric polytopes are more commonly known as regular polytopes. Before giving a precise definition of a regular polytope, let’s consider some familiar, low-dimensional examples.