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A Universal Triangulation for Flat Tori

Authors Francis Lazarus, Florent Tallerie

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ACM Subject Classification
  • Mathematics of computing → Geometric topology
  • Mathematics of computing → Discrete mathematics
  • Theory of computation → Computational geometry
Keywords
  • Triangulation
  • flat torus
  • isometric embedding

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Abstract

A result due to Burago and Zalgaller states that every orientable polyhedral surface, one that is obtained by gluing Euclidean polygons, has an isometric piecewise linear (PL) embedding into Euclidean space 𝔼³. A flat torus, resulting from the identification of the opposite sides of a Euclidean parallelogram, is a simple example of polyhedral surface. In a first part, we adapt the proof of Burago and Zalgaller, which is partially constructive, to produce PL isometric embeddings of flat tori. In practice, the resulting embeddings have a huge number of vertices, moreover distinct for every flat torus. In a second part, based on another construction of Zalgaller and on recent works by Arnoux et al., we exhibit a universal triangulation with 5974 triangles which can be embedded linearly on each triangle in order to realize the metric of any flat torus.

Cite As Get BibTex

Francis Lazarus and Florent Tallerie. A Universal Triangulation for Flat Tori. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 53:1-53:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.SoCG.2022.53

Author Details

Francis Lazarus
  • G-SCOP, CNRS, UGA, Grenoble, France
Florent Tallerie
  • G-SCOP, UGA, Grenoble, France

Funding

  • Lazarus, Francis: This author is partially supported by the French ANR projects GATO (ANR-16-CE40-0009-01) and MINMAX (ANR-19-CE40-0014) and the LabEx PERSYVAL-Lab (ANR-11-LABX-0025-01) funded by the French program Investissement d’avenir.

Acknowledgements

We warmly thank Alba Málaga, Pierre Arnoux and Samuel Lelièvre for sharing with us their constructions of flat tori and showing us how to cover their moduli space with these constructions. We are also grateful to the anonymous reviewers for their careful reading and suggestions.

References

  1. Pierre Arnoux, Samuel Lelièvre, and Alba Málaga. Diplotori: a family of polyhedral flat tori. In preparation, 2021. Google Scholar
  2. Thomas F. Banchoff. Geometry of the Hopf mapping and Pinkall’s tori of given conformal type. In Martin C. Tangora, editor, Computers in algebra, volume 111 of Lecture notes in pure and applied mathematics, pages 57-62. M. Dekker, 1988. Google Scholar
  3. Yuriy Dmitrievich Burago and Viktor Abramovich Zalgaller. Isometric piecewise-linear imbeddings of two-dimensional manifolds with a polyhedral metric into ℝ³. Algebra i analiz, 7(3):76-95, 1995. Transl. in St Petersburg Math. J. (7)3:369-385. Google Scholar
  4. Jin ichi Itoha and Liping Yuan. Acute triangulations of flat tori. European journal of combinatorics, 30:1-4, 2009. URL: https://doi.org/10.1016/j.ejc.2008.03.005.
  5. Nicolaas Kuiper. On C¹-isometric imbeddings. Indagationes Mathematicae, 17:545-555, 1955. Google Scholar
  6. John F. Nash. C¹-isometric imbeddings. Annals of Mathematics, 60(3):383-396, 1954. URL: https://doi.org/10.2307/1969840.
  7. Ulrich Pinkall. Hopf tori in S³. Inventiones mathematicae, 81(2):379-386, 1985. URL: https://doi.org/10.1007/BF01389060.
  8. Tanessi Quintanar. An explicit PL-embedding of the square flat torus into 𝔼³. Journal of Computational Geometry, 11(1):615-628, 2020. URL: https://doi.org/10.20382/jocg.v11i1a24.
  9. Takashi Tsuboi. On origami embeddings of flat tori. arXiv preprint, 2020. URL: http://arxiv.org/abs/2007.03434.
  10. V. A. Zalgaller. Some bendings of a long cylinder. Journal of Mathematical Sciences, 100(3):2228-2238, 2000. URL: https://doi.org/10.1007/s10958-000-0007-3.
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