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Hereditary Polytopes

  • Mark Mixer
  • Egon SchulteEmail author
  • Asia Ivić Weiss
Chapter
Part of the Fields Institute Communications book series (FIC, volume 70)

Abstract

Every regular polytope has the remarkable property that it inherits all symmetries of each of its facets. This property distinguishes a natural class of polytopes which are called hereditary. Regular polytopes are by definition hereditary, but the other polytopes in this class are interesting, have possible applications in modeling of structures, and have not been previously investigated. This paper establishes the basic theory of hereditary polytopes, focussing on the analysis and construction of hereditary polytopes with highly symmetric faces.

Keywords

Regular polytope Chiral polytope Extension of automorphisms 

Subject Classifications

51M20 52B15 

Notes

Acknowledgements

A substantial part of this article was written while we visited the Fields Institute for extended periods of time during the Thematic Program on Discrete Geometry and Applications in Fall 2011. We greatly appreciated the hospitality of the Fields Institute and are very grateful for the support we received. Mark Mixer was Fields Postdoctoral Fellow in Fall 2011. Egon Schulte was also supported by NSF-grant DMS–0856675, and Asia Ivić Weiss by NSERC. Finally, we wish to thank Peter McMullen and Barry Monson for helpful comments.

References

  1. 1.
    Bosma, W., Cannon, C., Playoust, C.: The Magma algebra system I: the user language. J. Symb. Comput. 24, 235–265 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Conder, M.: Regular maps and hypermaps of Euler characteristic − 1 to − 200. J. Comb. Theory Ser. B. 99, 455–459 (2009). With associated lists available at http://www.math.auckland.ac.nz/~conder/
  3. 3.
    Conder, M., Hubard, I., Pisanski, T.: Constructions for chiral polytopes. J. Lond. Math. Soc. (2) 77, 115–129 (2008)Google Scholar
  4. 4.
    Coxeter, H.S.M.: Regular skew polyhedra in 3 and 4 dimensions and their topological analogues. Proc. Lond. Math. Soc. (2) 43, 33–62 (1937) (Reprinted with amendments in Twelve Geometric Essays, Southern Illinois University Press (Carbondale, 1968), 76–105.)Google Scholar
  5. 5.
    Coxeter, H.S.M.: Regular and semi-regular polytopes, I, Math. Z. 46, 380–407 (1940). In: Sherk, F.A., McMullen, P., Thompson, A.C., Weiss, A.I. (eds.) Kaleidoscopes: Selected Writings of H.S.M. Coxeter, pp. 251–278. Wiley-Interscience, New York (1995)Google Scholar
  6. 6.
    Coxeter, H.S.M., Moser, W.O.J.: Generators and Relations for Discrete Groups. Volume 14 of Ergebnisse der Mathematik und ihrer Grenzgebiete (Results in Mathematics and Related Areas), 4th edn. Springer, Berlin (1980)Google Scholar
  7. 7.
    Danzer, L.: Regular Incidence-Complexes and Dimensionally Unbounded Sequences of Such, I. In: Rosenfeld, M., Zaks, J. (eds.) North-Holland Mathematics Studies, Convexity and Graph Theory, vol. 87, pp. 115–127. North-Holland, Amsterdam (1984)Google Scholar
  8. 8.
    D’Azevedo, A.B., Jones, G.A., Schulte, E.: Constructions of chiral polytopes of small rank. Can. J. Math. 63, 1254–1283 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Graver, J.E., Watkins, M.E.: Locally Finite, Planar, Edge-Transitive Graphs. Memoirs of the American Mathematical Society, vol. 126. American Mathematical Society, Providence (1997)Google Scholar
  10. 10.
    Grünbaum, B., Shephard, G.C.: Tilings and Patterns. Freeman & Co., New York (1987)zbMATHGoogle Scholar
  11. 11.
    Hartley, M.I.: The atlas of small regular abstract polytopes. Period. Math. Hung. 53, 149–156 (2006) (http://www.abstract-polytopes.com/atlas/)
  12. 12.
    Hubard, I.: Two-orbit polyhedra from groups. Eur. J. Comb. 31, 943–960 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Hubard, I., Schulte, E.: Two-orbit polytopes (in preparation)Google Scholar
  14. 14.
    Lučić, Z., Weiss, A.I.: Regular polyhedra in hyperbolic three-space, Coxeter-Festschrift (Part III). Mitt. Math. Sem. Giessen 165, 237–252 (1984)Google Scholar
  15. 15.
    Martini, H.: A hierarchical classification of Euclidean polytopes with regularity properties. In: Bisztriczky, T., McMullen, P., Schneider, R., Weiss, A.I. (eds.) Polytopes: Abstract, Convex and Computational, pp. 71–96. NATO ASI Series C, vol. 440. Kluwer, Dordrecht (1994)Google Scholar
  16. 16.
    McMullen, P., Schulte, E.: Abstract Regular Polytopes. Encyclopedia of Mathematics and Its Applications, vol. 92. Cambridge University Press, Cambridge (2002)Google Scholar
  17. 17.
    Monson, B., Schulte, E.: Semiregular polytopes and amalgamated C-groups. Adv. Math. 229, 2767–2791 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Orbanić, A., Pellicer, D., Weiss, A.I.: Map operations and k-orbit maps. J. Comb. Theory A 117, 411–429 (2012)CrossRefGoogle Scholar
  19. 19.
    Pellicer, D.: A construction of higher rank chiral polytopes. Discret. Math. 310, 1222–1237 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Schulte, E.: Extensions of regular complexes. In: Baker, C.A., Batten, L.M. (eds.) Finite Geometries. Lecture Notes Pure Applied Mathematics, vol. 103, pp. 289–305. Marcel Dekker, New York (1985)Google Scholar
  21. 21.
    Schulte, E.: Space-fillers of higher genus. J. Comb. Theory A 68, 438–453 (1994)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Schulte, E., Weiss, A.I.: Chirality and projective linear groups. Discret. Math. 131, 221–261 (1994)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Schulte, E., Weiss, A.I.: Problems on polytopes, their groups, and realizations. Period. Math. Hung. 53, 231–255 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Širan, J., Tucker, T.W., Watkins, M.: Realizing finite edge-transitive orientable maps. J. Graph Theory 37, 1–34 (2001)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Wentworth Institute of TechnologyBostonUSA
  2. 2.Department of MathematicsNortheastern UniversityBostonUSA
  3. 3.Department of Mathematics and StatisticsYork UniversityTorontoCanada

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