Odd or Even, Jitterbug Versus Grünbaum’s Double Polyhedra

Abstract

Coxeter, Longuet-Higgins & Miller (Coxeter et al. Uniform polyhedra. Philosophical Transactions of the Royal Society London 401–450, 1954) define uniform polyhedra to be vertex-transitive polyhedra with regular faces. They define a polyhedron to be a finite set of polygons such that each side of a polygon is a side of just one other polygon, such that no non-empty proper subset of the polygons has the same property. By a polygon they implicitly mean a polygon in 3-dimensional Euclidean space; these are allowed to be non-convex and to intersect each other. The Jitterbug transformation is a transformation that can be applied on uniform polyhedra in which the number of faces that meet in each vertex is even. Face-doubling, a method to generate new uniform polyhedra by doubling the faces of a known uniform polyhedron, can only be applied if there is at least one vertex in which an odd number of faces come together. This is what Grünbaum stated in his paper “New” Uniform Polyhedra (Grünbaum, Discrete Geometry: In Honor of W. Kuperberg’s 60th Birthday. Marcel Dekker, New York, 2003). So, for each uniform polyhedron, it seems that either the Jitterbug transformation or face-doubling applies. In this paper, I show that this is not always true. And that this fact leads to the discovery of a new uniform polyhedron.