Jackson Petty

Optimal monohedral tilings of hyperbolic surfaces

Abstract. The hexagon is the least-perimeter tile in the Euclidean plane for any given area. On hyperbolic surfaces, this ‘isoperimetric’ problem differs for every given area, as solutions do not scale. Cox conjectured that a regular $k$-gonal tile with $120$-degree angles is isoperimetric. For area $\pi/3$, the regular heptagon has $120$-degree angles and therefore tiles many hyperbolic surfaces. For other areas, we show the existence of many tiles but provide no conjectured optima. On closed hyperbolic surfaces, we verify via a reduction argument using cutting and pasting transformations and convex hulls that the regular $7$-gon is the optimal $n$-gonal tile of area $\pi/3$ for $3\leq n\leq 10$. However, for $n>10$, it is difficult to rule out non-convex $n$-gons that tile irregularly.