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Incidence-polytopes of type {6, 3, 3}

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Abstract

From the hyperbolic honeycomb {6, 3, 3} we derive a class of abstract polytopes whose cells are isomorphic to the toroidal maps {6, 3} b,c and vertex figures to tetrahedra. We give a criterion on the finiteness of these incidence-polytopes in terms of the group PSL ± (2, ℤ[ω]), leading, among other things, to the explicit recognition of the groups in some interesting special cases.

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References

  1. Bianchi, L., ‘Geometrische Darstellung der Gruppen linearer Substitutionen mit ganzen complexen Coefficienten nebst Anwendungen auf die Zahlentheorie’, Math. Ann. 38 (1891), 313–333.

    Article  MathSciNet  Google Scholar 

  2. Colbourn, C. J. and Ivić Weiss, A., ‘A Census of Regular 3-Polystroma Arising from Honeycombs’, Discrete Math. 50 (1984), 29–36.

    Article  MathSciNet  MATH  Google Scholar 

  3. Coxeter, H. S. M., Twelve Geometric Essays, Southern Illinois Univ. Press. Carbondale, Ill., 1968.

    MATH  Google Scholar 

  4. Coxeter, H. S. M., Twisted Honeycombs, Regional Conf. Series in Mathematics, No. 4, American Mathematical Society, Providence, R.I., 1970.

    MATH  Google Scholar 

  5. Coxeter, H. S. M., Regular Polytopes (3rd edn.), Dover, New York, 1973.

    MATH  Google Scholar 

  6. Coxeter, H. S. M. and Moser, W. O. J., Generators and Relations for Discrete Groups (4th ed.), Springer, Berlin, 1980.

    Book  MATH  Google Scholar 

  7. Danzer, L., ‘Regular Incidence-Complexes and Dimensionally Unbounded Sequence of Such, I’, Proc. Int. Conf. Convexity and Graph Theory, Israel, 1981.

  8. Danzer, L. and Schulte, E., ‘Reguläre Incidenzkomplexe I’, Geom. Dedicata 13 (1982), 295–308.

    Article  MathSciNet  MATH  Google Scholar 

  9. Dickson, L. E., Introduction to the Theory of Numbers, Dover, New York, 1957.

    MATH  Google Scholar 

  10. Grünbaum, B., Convex Polytopes, Wiley, London, 1967.

    MATH  Google Scholar 

  11. Grünbaum, B., ‘Regularity of Graphs, Complexes and Designs’, Coll. Int. C.N.R.S. No. 260, Problemeś Combinatorie et Théorie des Graphes, Orsay, 1977, pp. 191–197.

    Google Scholar 

  12. Liebmann, H., Nichteuklidische Geometrie, Leipzig, 1905.

  13. McMullen, P., ‘Combinatorially Regular Polytopes’, Mathematika 14 (1967), 142–150.

    Article  MathSciNet  MATH  Google Scholar 

  14. Schulte, E., ‘Reguläre Incidenzkomplexe II’, Geom. Dedicata 14 (1983), 33–56.

    MathSciNet  MATH  Google Scholar 

  15. Schulte, E., ‘Reguläre Incidenzkomplexe III’, Geom. Dedicata 14 (1983), 57–79.

    MathSciNet  MATH  Google Scholar 

  16. Schulte, E., ‘Regular Incidence-Polytopes with Euclidean or Toroidal Faces and VertexFigures’ (to appear).

  17. Tits, J., ‘A Local Approach to Buildings’, The Geometric Vein (C. Davis, B. Grübaum, and F. A. Sherk (eds)), Springer, New York, 1982, pp. 519–547.

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, York University, 4700 Keele Street, M3J IP3, North York, Ontario, Canada

    Asia Ivić Weiss

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  1. Asia Ivić Weiss
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Research supported by NSERC Canada Grant A8857.

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Weiss, A.I. Incidence-polytopes of type {6, 3, 3}. Geom Dedicata 20, 147–155 (1986). https://doi.org/10.1007/BF00164396

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