A New Family of Solids: The Infinite Kepler-Poinsot Polyhedra

Abstract

The so-called Platonic solids have fascinated mathematicians and artists for over 2000 years. It is astonishing that there are only five cases of regular polyhedra, that is, of polyhedra in which regular polygons form the same spatial angles between them in each vertex. In 1619, Kepler added the small and great stellated dodecahedron to this list, but he allowed intersecting faces. Poinsot did so too, in 1809, and discovered the great dodecahedron and great icosahedron. In 20th century, Coxeter and Petrie added three more regular polyhedra, using infinitely repeating elements, based on the truncated tetrahedron, the cube and the truncated octahedron. The principle of intersecting faces, typical for the Kepler-Poinsot solids, can be combined with the Coxeter-Petrie generalization to the infinite case. Thus, a new regular polyhedron was discovered, based on the cubohemioctahedron but without its square faces. Placed side by side and on top of each other, identical regular hexagons meet in each vertex, always with the same spatial angle. There are 8 of them in each vertex, and so it is not a compound of twice two polyhedra with 4 hexagons in each vertex. The dual of this {6, 8} polyhedron of infinite Kepler-Poinsot type is indeed a {8, 6} polyhedron of infinite Kepler-Poinsot type, if two overlapping squares are considered as one 8/4 octagonal star. Kepler-Poinsot solids are difficult to interpret, with their intersecting faces, and this infinite case is even more difficult to grasp. The present paper tries to solve this using open faces so that one can see through the solids.