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Certain hyperbolic regular polygonal tiles are isoperimetric

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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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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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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Keywords

  • Isoperimetric
  • Hyperbolic geometry
  • Tiling
  • Closed hyperbolic surfaces

Mathematics Subject Classification

  • 52C20
  • 51M09