Hexagonal Extensions of Toroidal Maps and Hypermaps

  • Maria Elisa Fernandes
  • Dimitri Leemans
  • Asia Ivić Weiss
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 234)


The rank 3 concept of a hypermap has recently been generalized to a higher rank structure in which hypermaps can be seen as “hyperfaces” but very few examples can be found in literature. We study finite rank 4 structures obtained by hexagonal extensions of toroidal hypermaps. Many new examples are produced that are regular or chiral, even when the extensions are polytopal. We also construct a new infinite family of finite nonlinear hexagonal extensions of the tetrahedron.


Regularity Chirality Thin geometries Hypermaps Abstract polytopes 

2000 Math Subj. Class:

52B11 20D06 



The authors would like to thank two anonymous referees for their numerous helpful and insightful comments. This research was supported by a Marsden grant (UOA1218) of the Royal Society of New Zealand, by NSERC and by the Portuguese Foundation for Science and Technology (FCT-Fundação para a Ciência e a Tecnologia), through CIDMA - Center for Research and Development in Mathematics and Applications, within project UID/MAT/04106/2013.


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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Maria Elisa Fernandes
    • 1
  • Dimitri Leemans
    • 2
    • 3
  • Asia Ivić Weiss
    • 4
  1. 1.Center for Research and Development in Mathematics and Applications, Department of MathematicsUniversity of AveiroAveiroPortugal
  2. 2.Department of MathematicsUniversity of AucklandAucklandNew Zealand
  3. 3.Département de Mathématique, C.P. 216 - Algèbre et CombinatoireUniversité Libre de BruxellesBrusselsBelgium
  4. 4.Department of Mathematics and StatisticsYork UniversityTorontoCanada

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