Skip to main content
SpringerLink
Log in

Self-Dual Regular 4-Polytopes and Their Petrie-Coxeter-Polyhedra

  • Published:
Results in Mathematics Aims and scope Submit manuscript

Abstract

Coxeter’s regular skew polyhedra in euclidean 4-space IE4 are intimately related to the self-dual regular 4-polytopes. The same holds for two of the three Petrie-Coxeter-polyhedra and the (self-dual) cubical tessellation in IE3.

In this paper we discuss the combinatorial Petrie-Coxeter-polyhedra associated with the self-dual regular 4-incidence-polytopes.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. H. S. M. Coxeter: Regular skew polyhedra in three and four dimensions and their topological analogues, in: Twelve geometric essays, Southern Illinois University Press, Carbondale, 1968, 75-105 (reprinted from: Proc. London Math. Soc. (2) 43 (1937), 33 - 62)

  2. H. S. M. Coxeter: Regular polytopes, 3rd edition, Dover, New York, 1973

    Google Scholar 

  3. H. S. M. Coxeter: A symmetrical arrangement of eleven hemi-icosahedra, Ann. Discrete Mathematics 20 (1984) 103–114

    MathSciNet  Google Scholar 

  4. L. Danzer and E. Schulte: Reguläre Inzidenzkomplexe 1, Geometriae dedicata 13 (1982), 295–308

    Article  MATH  MathSciNet  Google Scholar 

  5. B. Grünbaum: Regularity of graphs, complexes and designs, in: Coll. Int. C.N.R.S. No. 260, Problèmes Combinatoires et Théorie des Graphes, Orsay (1977), 191-197

  6. P. McMullen: Combinatorially regular polytopes, Mathematika 14 (1967), 142–150

    Article  MATH  MathSciNet  Google Scholar 

  7. P. McMullen and E. Schulte: Regular polytopes from twisted Coxeter-groups 1, 11, in preparation

  8. P. McMullen, E. Schulte and ]. M. Wills: Infinite series of combinatorially regular polyhedra in three-space, Geometriae dedicata, in print

  9. P. McMullen, Ch. Schulz and J. M. Wills: Equivelar polyhedral manifolds in IE3, Israel J. Math. 41 (1982), 331–346

    Article  MATH  MathSciNet  Google Scholar 

  10. E. Schulte: Reguläre Inzidenzkomplexe II, Geometriae dedicata 14 (1983), 33–56

    MATH  MathSciNet  Google Scholar 

  11. E. Schulte: Amalgamation of regular incidence-polytopes, Proc. London Math. Soc., in print

  12. E. Schulte and J. M. Wills: On Coxeter’s regular skew polyhedra, Discrete Math. 60 (1986), 253–262

    Article  MATH  MathSciNet  Google Scholar 

  13. H. Seifert and W. Threlfall: Lehrbuch der Topologie, Teubner, Leipzig, 1934

    Google Scholar 

  14. J. Tits: A local approach to buildings, in: The Geometric Vein (The Coxeter Festschrift), edit. Ch. Davis, B. Grünbaum and F. A. Sherk, Springer, Berlin, 1981

    Google Scholar 

  15. A. I. Weiss and Z. Lučić: Regular polyhedra in hyperbolic three- space, to appear

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University College London, Gower Street, London, WC1E 6BT, England

    P. McMullen

  2. Fachbereich Mathematik, Universität Dortmund, Postfach 50 05 00, D-4600, Dortmund, West Germany

    E. Schulte

Authors
  1. P. McMullen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. E. Schulte
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

McMullen, P., Schulte, E. Self-Dual Regular 4-Polytopes and Their Petrie-Coxeter-Polyhedra. Results. Math. 12, 366–375 (1987). https://doi.org/10.1007/BF03322401

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF03322401

Keywords

Search

Navigation