The Story of the 120-Cell, Volume 48, Number 1

O ne of the most beautiful objects in mathematics is the regular polytope in R4 whose boundary consists of 120 dodecahedral cells. This 120-cell is a rarity among rarities because it lives in three very special worlds. Its home is among the regular polytopes in R4, but it also lives in the remarkable sphere S3 and in the quaternions H. And if this is not enough, the 120-cell encodes the symmetry of the icosahedron and the structure of the Poincaré homology sphere. All these facts have been known since the 1930s, but the story can be told more elegantly in contemporary language, and it can be illustrated better than ever before with the help of computer graphics. Moreover, the new illustrations put the 120-cell in a context of current interest, the geometry of soap bubble configurations, by mapping it in a natural way into R3 (cover illustration). Telling the story in contemporary language has the danger that certain connections become “obvious”, and it is hard to understand how our mathematical ancestors could have overlooked them. However, there is no turning back; we cannot stop seeing the connections we see now, so the best thing to do is describe them as clearly as possible and recognise that our ancestors lacked our advantages. The story begins with the first encounters with the fourth dimension in the 1840s, becomes entangled with group theory in the 1850s, and interacts with topology around 1900. But to set the scene properly, we should review the regular polyhedra, because they are the origin of everything we are going to discuss. The Regular Polyhedra The five regular polyhedra existed before human history (for example, in the form of crystals and viruses), and they certainly made an early appearance in the history of mathematics. They are the climax of Euclid’s Elements. There are several proofs that these five polyhedra are the only regular ones: the classical proof enumerating which polygons can occur as faces and which angle sums are possible at a vertex, the topological proof showing that everything is controlled by the Euler characteristic, and the nice spherical geometry proof of Legendre. A less elegant proof, but one that generalises to higher dimensions, considers the ratio of edge length to the diameter of the circumscribed sphere and relates it to the corresponding ratio of the lower-dimensional vertex figure (the convex hull of the neighbouring vertices of a given vertex). Call this ratio ER. It is easy to prove that if Π is a polyhedron with p-gons as faces and if the polygon Π′ is its vertex figure, then