The role of convexity in defining regular polyhedra

It is almost twenty years since Branko Grünbaum lamented that the ‘original sin’ in the theory of polyhedra is that from Euclid onwards “the writers failed to define what are the ‘polyhedra’ among which they are finding the ‘regular’ ones” ([1, p. 43]). Various definitions of ‘regular’ can be found in the literature with a condition of convexity often included (e.g. [2, p. 301], [3, p. 77], [4, p. 47], [5, p. 435], [6, p. 16]). The condition of convexity is usually cited to exclude regular self-intersecting polyhedra, i.e. the Kepler-Poinsot polyhedra, such as the ‘great dodecahedron’ consisting of twelve intersecting pentagonal faces shown in Figure 1 with one face shaded. Richeson also notes ([4, pp. 47-48]) that, for a particular definition of ‘regular’, convexity is needed to exclude the ‘punched-in’ icosahedron shown in Figure 2.