Rigidity and Symmetry pp 97-116

Part of the Fields Institute Communications book series (FIC, volume 70) | Cite as

Variance Groups and the Structure of Mixed Polytopes

Chapter

Abstract

The natural mixing construction for abstract polytopes provides a way to build a minimal common cover of two regular or chiral polytopes. With the help of the chirality group of a polytope, it is often possible to determine when the mix of two chiral polytopes is still chiral. By generalizing the chirality group to a whole family of variance groups, we can explicitly describe the structure of the mix of two polytopes. We are also able to determine when the mix of two polytopes is invariant under other external symmetries, such as duality and Petrie duality.

Keywords

Abstract regular polytope Chiral polytope Self-dual polytope Chiral map Petrie dual External symmetry 

Subject Classifications

Primary 52B15 Secondary: 51M20 05C25 

References

  1. 1.
    Conder, M., Hubard, I., Pisanski, T.: Constructions for chiral polytopes. J. Lond. Math. Soc. (2) 77(1), 115–129 (2008). MR 2389920 (2009b:52031)Google Scholar
  2. 2.
    Coxeter, H.S.M., Moser, W.O.J.: Generators and relations for discrete groups. Ergebnisse der Mathematik und ihrer Grenzgebiete. Results in Mathematics and Related Areas, vol. 14, 4th edn. Springer, Berlin (1980). MR 562913 (81a:20001)Google Scholar
  3. 3.
    Cunningham, G.: Constructing self-dual chiral polytopes. Eur. J. Comb. 33(6), 1313–1323 (2012)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Cunningham, G.: Mixing chiral polytopes. J. Alg. Comb. 36(2), 263–277 (2012)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Cunningham, G.: Mixing regular convex polytopes. Disc. Math. 312(6), 763–771 (2012)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Cunningham, G.: Self-dual, self-petrie covers of regular polyhedra. Symmetry 4(1), 208–218 (2012)CrossRefMathSciNetGoogle Scholar
  7. 7.
    D’Azevedo, A.B., Jones, G., Nedela, R., Škoviera, M.: Chirality groups of maps and hypermaps. J. Algebraic Comb. 29(3), 337–355 (2009). MR 2496311 (2010g:05159)Google Scholar
  8. 8.
    D’Azevedo, A.B., Jones, G., Schulte, E.: Constructions of chiral polytopes of small rank. Canad. J. Math. 63(6), 1254–1283 (2011)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    D’Azevedo, A.B., Nedela, R.: Join and intersection of hypermaps. Acta Univ. M. Belii Ser. Math. 9, 13–28 (2001). MR 1935680 (2004f:05043)Google Scholar
  10. 10.
    Hubard, I., Orbanić, A., Ivić Weiss, A.: Monodromy groups and self-invariance. Canad. J. Math. 61(6), 1300–1324 (2009). MR 2588424 (2011a:52032)Google Scholar
  11. 11.
    Hubard, I., Weiss, A.I.: Self-duality of chiral polytopes. J. Comb. Theory Ser. A 111(1), 128–136 (2005). MR 2144859 (2006b:52011)Google Scholar
  12. 12.
    Jones, G.A., Thornton, J.S.: Operations on maps, and outer automorphisms. J. Comb. Theory Ser. B 35(2), 93–103 (1983). MR 733017 (85m:05036)Google Scholar
  13. 13.
    Jones, G.A., Poulton, A.: Maps admitting trialities but not dualities. Eur. J. Comb. 31(7), 1805–1818 (2010). MR 2673020 (2011m:20004)Google Scholar
  14. 14.
    Lang, S.: Algebra. Graduate Texts in Mathematics, vol. 211, 3rd edn. Springer, New York (2002). MR 1878556 (2003e:00003)Google Scholar
  15. 15.
    McMullen, P., Schulte, E.: Abstract regular polytopes. Encyclopedia of Mathematics and Its Applications, vol. 92. Cambridge University Press, Cambridge (2002). MR 1965665 (2004a:52020)Google Scholar
  16. 16.
    McMullen, P., Schulte, E.: The mix of a regular polytope with a face. Ann. Comb. 6(1), 77–86 (2002). MR 1923089 (2003h:52012)Google Scholar
  17. 17.
    Richter, R.B., Širán, J., Wang, Y.: Self-dual and self-petrie-dual regular maps. J. Graph Theory 69(2), 152–159 (2012)CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Schulte, E., Weiss, A.I.: Chiral polytopes. Applied Geometry and Discrete Mathematics. DIMACS Series in Discrete Mathematics and Theoretical Computer Science, vol. 4, pp. 493–516. American Mathematical Society, Providence (1991). MR 1116373 (92f:51018)Google Scholar
  19. 19.
    The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.7.4; 2014. (http://www.gap-system.org)
  20. 20.
    Wilson, S.E.: Parallel products in groups and maps. J. Algebra 167(3), 539–546 (1994). MR 1287058 (95d:20067)Google Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Massachusetts BostonBostonUSA

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