A combinatorial theory of Grünbaum's new regular polyhedra, Part II: Complete enumeration
- Andreas W. M. Dress
- … show all 1 hide
Rent the article at a discountRent now
* Final gross prices may vary according to local VAT.Get Access
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 spanE 3 and two further classes which are closed with respect to Petrie duality, each containing 3 individuals, all spanningE 3, 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 inE n is worked out in §9.
- Abels, H. andDress, A.,An algebraic version of a theorem of L. Bieberbach, concerning invariant subspaces of discrete isometry groups. Submitted to the J. Algebra.
- Bieberbach, L. (1910) Über die Bewegungsgruppen der Euklidischen Räume (Erste Abhandlung). Math. Ann. 70: pp. 297-336
- Brown, H., Bulow, R., Neubüser, J., Wonratschek, H., Zassenhaus, H. (1978) Crystallographic groups of four-dimensional space. Wiley, New York
- Dress, A. (1981) A combinatorial theory of Grünbaum's new regular polyhedra, Part I: Grünbaum's new regular polyhedra and their automorphism group. Aequationes Math. 23: pp. 252-264
- Grünbaum, B. (1977) Regular polyhedra—old and new. Aequationes Math. 16: pp. 1-20
- A combinatorial theory of Grünbaum's new regular polyhedra, Part II: Complete enumeration
Volume 29, Issue 1 , pp 222-243
- Cover Date
- Print ISSN
- Online ISSN
- Additional Links
- Primary 51M20
- Industry Sectors
- Author Affiliations
- 1. Universität Bielefeld, Fakultät für Mathematik, D-4800, Bielefeld, West Germany