A combinatorial theory of Grünbaum's new regular polyhedra, Part II: Complete enumeration
- Cite this article as:
- Dress, A.W.M. Aeq. Math. (1985) 29: 222. doi:10.1007/BF02189831
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The new regular polyhedra as defined by Branko Grünbaum in 1977 (cf. ) are completely enumerated. By means of a theorem of Bieberbach, concerning the existence of invariant affine subspaces for discrete affine isometry groups (cf. ,  or ) the standard crystallographic restrictions are established for the isometry groups of the non finite (Grünbaum-)polyhedra. Then, using an appropriate classification scheme which—compared with the similar, geometrically motivated scheme, used originally by Grünbaum—is suggested rather by the group theoretical investigations in , it turns out that the list of examples given in  is essentially complete except for one additional polyhedron.
So altogether—up to similarity—there are two classes of planar polyhedra, each consisting of 3 individuals and each class consisting of the Petrie duals of the other class, and there are ten classes of non planar polyhedra: two mutually Petrie dual classes of finite polyhedra, each consisting of 9 individuals, two mutually Petrie dual classes of infinite polyhedra which are contained between two parallel planes with each of those two classes consisting of three one-parameter families of polyhedra, two further mutually Petrie dual classes each of which consists of three one parameter families of polyhedra whose convex span is the whole 3-space, two further mutually Petrie dual classes consisting of three individuals each of which spanE3 and two further classes which are closed with respect to Petrie duality, each containing 3 individuals, all spanningE3, two of which are Petrie dual to each other, the remaining one being Petrie dual to itself.
In addition, a new classification scheme for regular polygons inEnis worked out in §9.