  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.
