aequationes mathematicae

, Volume 29, Issue 1, pp 222–243

A combinatorial theory of Grünbaum's new regular polyhedra, Part II: Complete enumeration

  • Andreas W. M. Dress
Research Papers

DOI: 10.1007/BF02189831

Cite this article as:
Dress, A.W.M. Aeq. Math. (1985) 29: 222. doi:10.1007/BF02189831


The new regular polyhedra as defined by Branko Grünbaum in 1977 (cf. [5]) are completely enumerated. By means of a theorem of Bieberbach, concerning the existence of invariant affine subspaces for discrete affine isometry groups (cf. [3], [2] or [1]) 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 [4], it turns out that the list of examples given in [5] 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.

AMS (1980) subject classification

Primary 51M20 51F15 

Copyright information

© Birkhäuser Verlag 1985

Authors and Affiliations

  • Andreas W. M. Dress
    • 1
  1. 1.Universität Bielefeld, Fakultät für MathematikBielefeldWest Germany