The Twist Operator on Maniplexes

  • Ian Douglas
  • Isabel Hubard
  • Daniel Pellicer
  • Steve Wilson
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 234)


Maniplexes are combinatorial objects that generalize, simultaneously, maps on surfaces and abstract polytopes. We are interested on studying highly symmetric maniplexes, particularly those having maximal ‘rotational’ symmetry. This paper introduces an operation on polytopes and maniplexes which, in its simplest form, can be interpreted as twisting the connection between facets. This is first described in detail in dimension 4 and then generalized to higher dimensions. Since the twist on a maniplex preserves all the orientation preserving symmetries of the original maniplex, we apply the operation to reflexible maniplexes, to attack the problem of finding chiral polytopes in higher dimensions.


Graph Automorphism group Symmetry Polytope Maniplex Map Flag Transitivity Rotary Reflexible Chiral 



We gratefully acknowledge financial support of the PAPIIT-DGAPA, under grant IN107015, and of CONACyT, under grant 166951. The completion of this work was done while the second author was on sabbatical at the Laboratoire d’Informatique de l’École Polytechnique. She thanks LIX and Vincent Pilaud for their hospitality, as well as the program PASPA-DGAPA and the UNAM for the support for this sabbatical stay.


  1. 1.
    H.S.M. Coxeter, Regular Polytopes, 3rd edn. (Dover, New York, 1973)zbMATHGoogle Scholar
  2. 2.
    H.S.M. Coxeter, The edges and faces of a 4-dimensional polytope, in Proceedings of the Eleventh Southeastern Conference on Combinatorics, Graph Theory and Computing (Florida Atlantic University, Boca Raton, FL, 1980), Vol. I. Congr. Numer. 28 (1980), pp. 309–334Google Scholar
  3. 3.
    G. Cunningham, M. Del Rio-Francos, I. Hubard, M. Toledo, Symmetry type graphs of polytopes and maniplexes. Ann. Comb. 19, 243–268 (2015)MathSciNetCrossRefGoogle Scholar
  4. 4.
    I. Douglas, Operators on Maniplexes, NAU Thesis series (2012)Google Scholar
  5. 5.
    L. Danzer, Regular incidence-complexes and dimensionally unbounded sequences of such, I, In Convexity and Graph Theory (Jerusalem 1981), North-Holland Math. Stud. vol. 87 (North-Holland, Amsterdam, 1984), pp. 115–127Google Scholar
  6. 6.
    M.E. Fernandes, D. Leemans, A.I. Weiss, Highly symmetric hypertopes. Aequationes Math. 90, 1045–1067 (2016)MathSciNetCrossRefGoogle Scholar
  7. 7.
    J. Garza-Vargas, I. Hubard, Politopality of Maniplexes. Discrete mathematics (2018)Google Scholar
  8. 8.
    M.I. Hartley, An atlas of small regular polytopes. Period. Math. Hung.53, pp. 149–156 (2006) (Full atlas online at
  9. 9.
    I. Hubard, A. Orbanić, A.I. Weiss, Monodromy groups and self-invariance. Canad. J. Math. 61, 1300–1324 (2009)MathSciNetCrossRefGoogle Scholar
  10. 10.
    H. Koike, D. Pellicer, M. Raggi, S. Wilson, Flag bicolorings, pseudo-orientations, and double covers of maps (submitted)Google Scholar
  11. 11.
    S. Krughoff, Rotary maniplexes with one and two facets, NAU Thesis series (2012)Google Scholar
  12. 12.
    P. McMullen, E. Schulte, Abstract Regular Polytopes, 1st edn. (Cambridge University Press, Cambridge, 2002)CrossRefGoogle Scholar
  13. 13.
    B. Monson, D. Pellicer, G. Williams, Mixing and monodromy of abstract polytopes. Trans. Am. Math. Soc. 366, 2651–2681 (2014)MathSciNetCrossRefGoogle Scholar
  14. 14.
    D. Pellicer, A construction of higher rank chiral polytopes. Discrete Math. 310, 1222–1237 (2010)MathSciNetCrossRefGoogle Scholar
  15. 15.
    D. Pellicer, Developments and open problems on chiral polytopes. Ars Math. Contemp. 5, 333–354 (2012)MathSciNetzbMATHGoogle Scholar
  16. 16.
    E. Schulte, Reguläre Inzidenzkomplexe, Universität Dortmund Dissertation (1980)Google Scholar
  17. 17.
    E. Schulte, Symmetry of polytopes and polyhedra, in Handbook of Discrete and Computational Geometry (CRC Press Ser. Discrete Math. Appl., CRC, Boca Raton, FL, 1997)Google Scholar
  18. 18.
    A. Vince, Combinatorial maps. J. Combin. Theory, Ser. B 34 (1983) 1-21MathSciNetCrossRefGoogle Scholar
  19. 19.
    S. Wilson, Maniplexes: part 1: maps, polytopes, symmetry and operators. Symmetry 4(2), 265–275 (2012)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Ian Douglas
    • 1
  • Isabel Hubard
    • 2
  • Daniel Pellicer
    • 3
  • Steve Wilson
    • 4
  1. 1.TucsonUSA
  2. 2.Instituto de MatemáticasUniversidad Nacional Autónoma de México Circuito ExteriorMexico D.F.Mexico
  3. 3.Centro de Ciencias Matemáticas, Universidad Nacional Autónoma de MéxicoMoreliaMexico
  4. 4.Department of Mathematics and StatisticsNorthern Arizona UniversityFlagstaffUSA

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