Regular Polytopes and Fold-Outs

Like many cultures before and since, the ancient Greeks were fascinated by polygons, planar figures bounded by line segments. Familiar polygons like squares and equilateral triangles and regular hexagons appear in all sorts of planar designs and in architectural constructions. These basic shapes combine with other polygons to form repeating patterns in the plane, and they combine in space to form three-dimensional polyhedral figures such as cubes and pyramids.

Of particular interest are the regular polygons and regular polyhedra, objects with the maximum amount of symmetry possible in their respective spaces. A "regular" polygon or polyhedron looks exactly the same at every vertex, a very strong restriction. In the plane, there are infinitely many regular polygons, each having a different number of sides. But in three-dimensional space, there are only five regular polyhedra. By the middle of the nineteenth century, geometers realized that there could be regular figures in dimensions four and higher, and they wondered how many and of what sorts. The race was on to find the answer to this challenging question, and after a few false starts, several mathematicians each claimed to be the first to solve the problem. The resolution of the dispute surprised everyone involved, as we will see by the end of a this chapter.