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Realising Equivelar Toroids of Type \(\{4,4\}\)

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Abstract

By an equivelar toroid we understand a map satisfying the requirements of an abstract polyhedron on the 2-torus where all faces have a fixed number p of edges, and all vertices belong to a fixed number q of edges. We establish that if every regular toroid of type \(\{4,4\}\) admits a faithful and symmetric realisation in a metric space \(\mathcal {S}\) then every equivelar toroid of type \(\{4,4\}\) admits a faithful and symmetric realisation in \(\mathcal {S}\). We exemplify this by showing that every equivelar toroid of type \(\{4,4\}\) admits faithful and symmetric realisations in the 3-sphere and in 3-dimensional projective space.

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Acknowledgments

The authors were partially supported by PAPIIT-UNAM under project IN112512, and by CONACyT under project 166951. We also thank the anonymous referees for useful suggestions of improvements.

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Authors and Affiliations

  1. Instituto de Matemáticas, Universidad Nacional Autónoma de México (UNAM), Mexico City, Mexico

    Javier Bracho & Isabel Hubard

  2. Centro de Ciencias Matemáticas, Universidad Nacional Autónoma de México (UNAM), Morelia, Mexico

    Daniel Pellicer

Authors
  1. Javier Bracho
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  2. Isabel Hubard
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  3. Daniel Pellicer
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Correspondence to Daniel Pellicer.

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Bracho, J., Hubard, I. & Pellicer, D. Realising Equivelar Toroids of Type \(\{4,4\}\) . Discrete Comput Geom 55, 934–954 (2016). https://doi.org/10.1007/s00454-016-9775-5

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  • DOI: https://doi.org/10.1007/s00454-016-9775-5

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