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Chiral Polyhedra in 3-Dimensional Geometries and from a Petrie–Coxeter Construction

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Abstract

We study chiral polyhedra in 3-dimensional geometries (Euclidean, hyperbolic, and projective) in a unified manner. This extends to hyperbolic and projective spaces some structural results in the classification of chiral polyhedra in Euclidean 3-space given in 2005 by Schulte. Then, we describe a way to produce examples with helical faces based on a classic Petrie–Coxeter construction, yielding a new family in \(\mathbb {S}^3\) which is described exhaustively.

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Acknowledgements

The authors thank the financial support of projects PAPIIT-UNAM: IN-100518 and IN-109218. They are also grateful to the referees, whose suggestions improved the final version of this paper.

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Authors and Affiliations

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

    Javier Bracho & Isabel Hubard

  2. Centro de Ciencias Matemáticas, Universidad Nacional Autónoma de México, Morelia, México

    Daniel Pellicer

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

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Bracho, J., Hubard, I. & Pellicer, D. Chiral Polyhedra in 3-Dimensional Geometries and from a Petrie–Coxeter Construction. Discrete Comput Geom 66, 1025–1052 (2021). https://doi.org/10.1007/s00454-021-00317-0

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  • DOI: https://doi.org/10.1007/s00454-021-00317-0

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