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aequationes mathematicae

, Volume 29, Issue 1, pp 222–243 | Cite as

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

  • Andreas W. M. Dress
Research Papers

Abstract

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 inE n is worked out in §9.

AMS (1980) subject classification

Primary 51M20 51F15 

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References

  1. [1]
    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.Google Scholar
  2. [2]
    Bieberbach, L.,Über die Bewegungsgruppen der Euklidischen Räume (Erste Abhandlung). Math. Ann.70 (1910), 297–336.Google Scholar
  3. [3]
    Brown, H., Bulow, R., Neubüser, J., Wonratschek, H., andZassenhaus, H.,Crystallographic groups of four-dimensional space. Wiley, New York, 1978.Google Scholar
  4. [4]
    Dress, A.,A combinatorial theory of Grünbaum's new regular polyhedra, Part I: Grünbaum's new regular polyhedra and their automorphism group. Aequationes Math.23 (1981), 252–264.Google Scholar
  5. [5]
    Grünbaum, B.,Regular polyhedra—old and new. Aequationes Math.16 (1977), 1–20.Google Scholar

Copyright information

© Birkhäuser Verlag 1985

Authors and Affiliations

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

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