Regular Polyhedral Clones

Abstract

For any polyhedron (3-polytope) P in ordinary three-dimensional space let S(P) denote its surface, that is, the union of its (closed) 2-faces. If P1 and P2 are given polyhedra, they are said to be clones of each other if there exists an isometry which maps S(P1) onto S(P2). In practical terms, this means that if one makes a paper model of one of the polyhedra, say P1 then, ignoring the ‘creases’ corresponding to the edges of P1, S(P1) can be changed into S(P2) without any stretching or cutting.