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Combinatorica

, Volume 12, Issue 2, pp 203–220 | Cite as

Regular polytopes of type {4,4,3} and {4,4,4}

  • P. McMullen
  • E. Schulte
Article

Abstract

Abstract regular polytopes generalize the classical concept of a regular polytope and regular tessellation to more complicated combinatorial structures with a distinctive geometrical and topological flavour. In this paper the authors give an almost complete classification of the (universal) locally toroidal regular 4-polytopes of Schläfli types {4,4,3} and {4,4,4}.

AMS subject classification (1991)

51 M 20 52 B 15 

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References

  1. [1]
    N. Bourbaki:Groupes et algebres de Lie, Ch. 4–6, Actu. Sci. Ind., Hermann, Paris, 1968.zbMATHGoogle Scholar
  2. [2]
    F. Buekenhout: Diagrams for geometries and groups,J. Comb. Theory A 27 (1979), 121–151.MathSciNetCrossRefGoogle Scholar
  3. [3]
    C. J. Colbourn, andA. I. Weiss: A census of regular 3-polystroma arising from honeycombs,Discrete Mathematics 50 (1984), 29–36.MathSciNetCrossRefGoogle Scholar
  4. [4]
    H. S. M. Coxeter: Regular skew polyhedra in 3 and 4 dimensions and their topological analogues,Proc. London Math. Soc. (2)43 (1937), 33–62. (reprinted in [6])MathSciNetzbMATHGoogle Scholar
  5. [5]
    H. S. M. Coxeter: Groups generated by unitary reflections of period two,Canadian J. Math. 9 (1957), 243–272.MathSciNetCrossRefGoogle Scholar
  6. [6]
    H. S. M. Coxeter:Twelve geometric essays, Southern Illinois University Press, Carbondale, (1968), 199–214.zbMATHGoogle Scholar
  7. [7]
    H. S. M. Coxeter:Regular polytopes, 3rd edition, Dover, New York, (1973).zbMATHGoogle Scholar
  8. [8]
    H. S. M. Coxeter, andW. O. J. Moser:Generators and relations for discrete groups, 4th edition, Springer, Berlin, (1980).CrossRefGoogle Scholar
  9. [9]
    H. S. M. Coxeter andG. C. Shephard: Regular 3-complexes with toroidal cells,J. Comb. Theory B22 (1977), 131–138.MathSciNetCrossRefGoogle Scholar
  10. [10]
    L. Danzer, andE. Schulte: Reguläre Inzidenzkomplexe I,Geometriae Dedicata 13 (1982), 295–308.MathSciNetCrossRefGoogle Scholar
  11. [11]
    A. W. M. Dress: Regular polytopes and equivariant tessellations from a combinatorial point of view,Algebraic Topology (Göttingen 1984), Lecture Notes in Mathematics 1172, Springer, (1985), 56–72.Google Scholar
  12. [12]
    B. Grünbaum: Regularity of graphs, complexes and designs, in: Problèmes combinatories et théorie des graphes, Coll. Int. CNRS No.260, Orsay, 1977, 191–197.Google Scholar
  13. [13]
    P. McMullen: Combinatorially regular polytopes,Mathematika 14 (1967), 142–150.MathSciNetCrossRefGoogle Scholar
  14. [14]
    P. McMullen: Realizations of regular polytopes,Aequationes Math. 37 (1989), 38–56.MathSciNetCrossRefGoogle Scholar
  15. [15]
    P. McMullen, andE. Schulte: Constructions of regular polytopes,J. Comb. Theory, A53 (1990) 1–28.MathSciNetCrossRefGoogle Scholar
  16. [16]
    P. McMullen, andE. Schulte: Regular polytopes from twisted Coxeter groups,Mathematische Zeitschrift 201 (1989), 209–226.MathSciNetCrossRefGoogle Scholar
  17. [17]
    P. McMullen, andE. Schulte: Regular polytopes from twisted Coxeter groups and unitary reflexion groups,Advances in Mathematics 82 (1990) 35–87.MathSciNetCrossRefGoogle Scholar
  18. [18]
    E. Schulte: Reguläre Inzidenzkomplexe II.Geometriae Dedicata 14 (1983), 33–56.MathSciNetzbMATHGoogle Scholar
  19. [19]
    E. Schulte: Regular incidence-polytopes with Euclidean or toroidal faces and vertexfigures,J. Comb. Theory A 40 (1985), 305–330.CrossRefGoogle Scholar
  20. [20]
    E. Schulte: Amalgamations of regular incidence-polytopes,Proc. London Math. Soc. (3)56 (1988), 303–328.MathSciNetCrossRefGoogle Scholar
  21. [21]
    G. C. Shephard, andJ. A. Todd: Finite unitary reflection groups,Canadian J. Math. 6 (1954), 274–304.MathSciNetCrossRefGoogle Scholar
  22. [22]
    J. Tits: A local approach to buildings, in:The geometric vein (The Coxeter-Festschrift), edit. by Ch. Davis, B. Grünbaum and F. A. Sherk, Springer, Berlin, 1981, 519–547.CrossRefGoogle Scholar
  23. [23]
    A. I. Weiss: Incidence-polytopes of type {6, 3, 3},Geometriae Dedicata 20 (1986), 147–155.MathSciNetCrossRefGoogle Scholar
  24. [24]
    A. I. Weiss: Incidence-polytopes with toroidal cells,Discrete Computational Geometry 4 (1989), 55–73.MathSciNetCrossRefGoogle Scholar
  25. [25]
    A. I. Weiss: Some infinite families of finite incidence-polytopes,J. Combinatorial Theory A55 (1990), 60–73.MathSciNetCrossRefGoogle Scholar

Added in proof

  1. [26]
    P. McMullen, andE. Schulte: Twisted groups and locally toroidal regular polytopes, in preparation.Google Scholar

Copyright information

© Akadémiai Kiadó 1992

Authors and Affiliations

  • P. McMullen
    • 1
  • E. Schulte
    • 2
  1. 1.Department of MathematicsUniversity College LondonLondonEngland
  2. 2.Department of MathematicsCambridgeUSA

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