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Certain hyperbolic regular polygonal tiles are isoperimetric

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Abstract

The hexagon is the least-perimeter tile in the Euclidean plane. On hyperbolic surfaces, the isoperimetric problem differs for every given area. Cox conjectured that a regular k-gonal tile with 120-degree angles is isoperimetric for its area. We prove his conjecture and more.

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References

  1. Bezdek, K.: Ein elementarer Beweis für die isoperimetrische Ungleichung in der Euklidischen und hyperbolischen Ebene. Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 27, 107–112 (1984)

    MathSciNet  MATH  Google Scholar 

  2. Carroll, C., Jacob, A., Quinn, C., Walters, R.: On generalizing the Honeycomb theorem to compact hyperbolic manifolds and the sphere. Report of SMALL Geometry Group’06, Williams College (Aug. 2006). https://sites.williams.edu/Morgan/ongeneralizing-the-honeycomb-theorem-to-compact-hyperbolic-manifolds-and-thesphere/

  3. Choe, J.: On the existence and regularity of fundamental domains with least boundary area. J. Differ. Geom. 29(3), 623–663 (1989). https://doi.org/10.4310/jdg/1214443065

    Article  MathSciNet  MATH  Google Scholar 

  4. Cox, C.: The Honeycomb problem on hyperbolic surfaces (Aug. 2005). https://sites.williams.edu/Morgan/the-honeycomb-problem-on-hyperbolic-surfaces/

  5. Cox, C.: The Honeycomb problem on hyperbolic surfaces (July 2011). https://sites.williams.edu/Morgan/the-honeycomb-problem-on-hyperbolic-surfaces/

  6. Di Giosia, L., Habib, J., Hirsch, J., Kenigsberg, L., Li, K., Pittman, D., Petty, J., Xue, C., Zhu, W.: Optimal monohedral tilings of hyperbolic surfaces. arXiv e-prints arXiv:1911.04476 [math.MG] (Nov. 2019)

  7. Edmonds, A.L., Ewing, J.H., Kulkarni, R.S.: Regular tessellations of surfaces and (p, q, 2)-triangle groups. Ann. Math. 116(1), 113–132 (1982)

    Article  MathSciNet  Google Scholar 

  8. Edmonds, A.L., Ewing, J.H., Kulkarni, R.S.: Torsion free subgroups of Fuchsian groups and tessellations of surfaces. Bull. Am. Math. Soc. 6(3), 456–458 (1982). https://doi.org/10.1090/S0273-0979-1982-15014-5

    Article  MathSciNet  MATH  Google Scholar 

  9. Edmonds, A.L., Ewing, J.H., Kulkarni, R.S.: Torsion free subgroups of Fuchsian groups and tessellations of surfaces. Invent. Math. 69(3), 331–346 (1982). https://doi.org/10.1007/BF01389358

    Article  MathSciNet  MATH  Google Scholar 

  10. Hales, T.C.: The Honeycomb conjecture. Discrete Comput. Geom. 25(1), 1–22 (2001). https://doi.org/10.1007/s004540010071

    Article  MathSciNet  MATH  Google Scholar 

  11. Klein, F.: Ueber die Transformation siebenter Ordnung der elliptischen Functionen. Math. Ann. 4, 428–471 (1878). https://gdz.sub.uni-goettingen.de/download/pdf/PPN235181684_0014/PPN235181684_0014.pdf. Trans. by Silvio Levy as “On the order-seven transformation of elliptic functions.” In: The Eightfold Way: The Beauty of Klein’s Quartic Curve. Vol. 35 (1998). Chap. 9, pp. 287–331. http://library.msri.org/books/Book35/files/klein.pdf

  12. Šešum, V.: The Honeycomb Problem on Hyperbolic Surfaces. Undergraduate Thesis. Williams College (May 2006). https://sites.williams.edu/Morgan/the-honeycombproblem-on-hyperbolic-surfaces-by-vojislav-sesum/

  13. Tóth, L.F.: Regular Figures. Vol. 48. International Series of Monographs on Pure and Applied Mathematics. The Macmillan Company, New York (1964)

    Google Scholar 

  14. Tóth, L.F.: Über das kürzeste Kurvennetz das eine Kugeloberfläche in flächengleiche konvexe Teil zerlegt. Mat. Term. tud. Ertesitö 62, 349–354 (1943)

    MATH  Google Scholar 

  15. Tóth, L.F.: What the bees know and what they do not know. Bull. Am. Math. Soc. 70(4), 468–481 (1964)

    Article  MathSciNet  Google Scholar 

  16. University of Glasgow. Hyperbolic Area—Heron’s Formula. https://www.maths.gla.ac.uk/wws/cabripages/hyperbolic/harea2.html

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Acknowledgements

This work is a product of the 2019 Summer Undergraduate Mathematics Research program at Yale (SUMRY) under the guidance of Frank Morgan of Williams College. The authors greatly thank Morgan for his help and insight over the many weeks spent researching and writing this paper. We thank the Young Mathematicians Conference (YMC) and Yale for supporting our trip to present at the 2019 YMC in Columbus, Ohio.

Funding

The research was supported by 2019 Summer Undergraduate Mathematics Research program at Yale (SUMRY).

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  1. Department of Mathematics, Yale University, New Haven, CT, 06510, USA

    Jack Hirsch, Kevin Li, Jackson Petty & Christopher Xue

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  1. Jack Hirsch
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  2. Kevin Li
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  3. Jackson Petty
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  4. Christopher Xue
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Correspondence to Kevin Li.

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Hirsch, J., Li, K., Petty, J. et al. Certain hyperbolic regular polygonal tiles are isoperimetric. Geom Dedicata 214, 65–77 (2021). https://doi.org/10.1007/s10711-021-00605-2

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