Glasnik Matematicki, Vol. 47, No. 2 (2012), 421-430.

ON THE SIZE OF EQUIFACETTED SEMI-REGULAR POLYTOPES

Tomaž Pisanski, Egon Schulte and Asia Ivić Weiss

Faculty of Mathematics and Physics, University of Ljubljana, Ljubljana, Slovenia

Department of Mathematics, Northeastern University, Boston, Massachusetts, USA, 02115

Department of Mathematics and Statistics, York University, Toronto, ON, Canada M3J 1P3


Abstract.   Unlike the situation in the classical theory of convex polytopes, there is a wealth of semi-regular abstract polytopes, including interesting examples exhibiting some unexpected phenomena. We prove that even an equifacetted semi-regular abstract polytope can have an arbitrary large number of flag orbits or face orbits under its combinatorial automorphism group.

2010 Mathematics Subject Classification.   51M20, 52B15.

Key words and phrases.   Semi-regular polytope, uniform polytope, Archimedean solid, abstract polytope.


Full text (PDF) (free access)

DOI: 10.3336/gm.47.2.15


References:

  1. G. Blind and R. Blind, The semiregular polytopes, Comment. Math. Helv. 66 (1991), 150-154.
    MathSciNet     CrossRef

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

  3. H. S. M. Coxeter, Regular and semi-regular polytopes, I, Math. Z. 46 (1940), 380-407. (In Kaleidoscopes: Selected writings of H.S.M. Coxeter (eds. F. A. Sherk, P. McMullen, A. C. Thompson and A. I. Weiss), Wiley-Interscience, New York, etc., 1995, 251-278.)
    MathSciNet     CrossRef

  4. H. S. M. Coxeter, Regular and semiregular polytopes, II, Math. Z. 188 (1985), 559-591. (In Kaleidoscopes: Selected writings of H. S. M. Coxeter (eds. F. A. Sherk, P. McMullen, A. C. Thompson and A. I. Weiss), Wiley-Interscience, New York, etc., 1995, 279-311.)
    MathSciNet     CrossRef

  5. H. S. M. Coxeter, M. S. Longuet-Higgins and J. C. P. Miller, Uniform polyhedra, Phil. Trans. Roy. Soc. London Ser. A 246 (1954), 401-450.
    MathSciNet     CrossRef

  6. L. Danzer, Regular incidence-complexes and dimensionally unbounded sequences of such. I, in: Convexity and graph theory (Jerusalem 1981), North-Holland, Amsterdam, 1984, 115-127.
    MathSciNet     CrossRef

  7. B. Grünbaum, Regularity of graphs, complexes and designs, in: Problèmes combinatoires et théorie des graphes, Colloq. Internat. CNRS 260 (1978), 191-197.
    MathSciNet    

  8. T. Gosset, On the regular and semiregular figures in spaces of n dimensions, Messenger of Mathematics 29 (1900), 43-48.

  9. I. Hubard, Two-orbit polyhedra from groups, European J. Combin. 31 (2010), 943-960.
    MathSciNet     CrossRef

  10. N. W. Johnson, Uniform polytopes, book manuscript, in preparation.

  11. H. Martini, A hierarchical classification of euclidean polytopes with regularity properties, in: Polytopes: abstract, convex and computational (eds. T. Bisztriczky, P. McMullen, R. Schneider and A. Ivić Weiss), Kluwer, Dordrecht, 1994, 71-96.
    MathSciNet     CrossRef

  12. P. McMullen and E. Schulte, Abstract regular polytopes, Cambridge University Press, Cambridge, 2002.
    MathSciNet     CrossRef

  13. B. Monson and E. Schulte, Semiregular polytopes and amalgamated C-groups, Adv. Math. 229 (2012), 27672791.
    MathSciNet     CrossRef

  14. D. Pellicer, A construction of higher rank chiral polytopes, Discrete Math. 310 (2010), 1222-1237.
    MathSciNet     CrossRef

  15. E. Schulte, Extensions of regular complexes, in: Finite geometries (eds. C.A. Baker and L.M. Batten), Marcel Dekker, New York, 1985, 289-305.
    MathSciNet    

  16. E. Schulte and A. I. Weiss, Chirality and projective linear groups, Discrete Math. 131 (1994), 221-261.
    MathSciNet     CrossRef

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