Certain Hyperbolic Regular Polygonal Tiles are Isoperimetric

TL;DR

This paper proves Cox's conjecture that certain regular hyperbolic polygons with 120-degree angles are optimal tiles for enclosing a given area with minimal perimeter, advancing understanding of hyperbolic isoperimetric problems.

Contribution

The paper confirms Cox's conjecture and extends the results to a broader class of hyperbolic regular polygons, establishing their isoperimetric properties.

Findings

Regular hyperbolic polygons with 120-degree angles are isoperimetric for their area.

Proof of Cox's conjecture on hyperbolic polygonal tiles.

Extension of isoperimetric results to more hyperbolic polygons.

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.