Polyhedra with Hollow Faces

Abstract

The Original Sin in the theory of polyhedra goes back to Euclid, and through Kepler, Poinsot, Cauchy and many others continues to afflict all the work on this topic (including that of the present author). It arises from the fact that the traditional usage of the term “regular polyhedra” was, and is, contrary to syntax and to logic: the words seem to imply that we are dealing, among the objects we call “polyhedra”, with those special ones that deserve to be called “regular”. But at each stage— Euclid, Kepler, Poinsot, Hess, Brückner,…—the writers failed to define what are the “polyhedra” among which they are finding the “regular” ones. True, we now know what are the convex polyhedra, which we think are the polyhedra Euclid had in mind; hence there is no stigma attached to the use of a term like “regular convex polyhedron”. But where in the literature do we find acceptable definitions of polyhedra that could be specialized to give the “regular Kepler-Poinsot polyhedra” ? For these, a better expression would be to say that they are “regularpolyhedra”—a distinct kind of objects, constructed according to more or less explicit procedures, and without any connection to what the separate parts of that ungainly word may mean.

Research supported in part by NSF grants DMS-9008813 and DMS-9300657. Comments by Heidi Burgiel on an earlier version of this paper are acknowledged with thanks.