Skip to main content
SpringerLink
Log in

Variance Groups and the Structure of Mixed Polytopes

  • Chapter
  • First Online:
Rigidity and Symmetry

Part of the book series: Fields Institute Communications ((FIC,volume 70))

  • 948 Accesses

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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  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. 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. Cunningham, G.: Constructing self-dual chiral polytopes. Eur. J. Comb. 33(6), 1313–1323 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  4. Cunningham, G.: Mixing chiral polytopes. J. Alg. Comb. 36(2), 263–277 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  5. Cunningham, G.: Mixing regular convex polytopes. Disc. Math. 312(6), 763–771 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  6. Cunningham, G.: Self-dual, self-petrie covers of regular polyhedra. Symmetry 4(1), 208–218 (2012)

    Article  MathSciNet  Google Scholar 

  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. D’Azevedo, A.B., Jones, G., Schulte, E.: Constructions of chiral polytopes of small rank. Canad. J. Math. 63(6), 1254–1283 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  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. 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. 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. 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. 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. Lang, S.: Algebra. Graduate Texts in Mathematics, vol. 211, 3rd edn. Springer, New York (2002). MR 1878556 (2003e:00003)

    Google Scholar 

  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. 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. Richter, R.B., Širán, J., Wang, Y.: Self-dual and self-petrie-dual regular maps. J. Graph Theory 69(2), 152–159 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  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. The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.7.4; 2014. (http://www.gap-system.org)

  20. Wilson, S.E.: Parallel products in groups and maps. J. Algebra 167(3), 539–546 (1994). MR 1287058 (95d:20067)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Massachusetts Boston, Boston, MA, USA

    Gabe Cunningham

Authors
  1. Gabe Cunningham
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gabe Cunningham .

Editor information

Editors and Affiliations

  1. Department of Mathematics, Cornell University, Ithaca, New York, USA

    Robert Connelly

  2. Department of Mathematics and Statistics, York University, Toronto, Ontario, Canada

    Asia Ivić Weiss

  3. Department of Mathematics & Statistics, York University, Toronto, Ontario, Canada

    Walter Whiteley

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer Science+Business Media New York

About this chapter

Cite this chapter

Cunningham, G. (2014). Variance Groups and the Structure of Mixed Polytopes. In: Connelly, R., Ivić Weiss, A., Whiteley, W. (eds) Rigidity and Symmetry. Fields Institute Communications, vol 70. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-0781-6_6

Download citation

Publish with us

Policies and ethics

Search

Navigation