= Tessellation
A tessellation is a covering by tiles with pairwise disjoint interiors; a locally finite tessellation meets any compact set in only finitely many tiles. In a tessellation by congruent compact hyperbolic polygons, the adjacency graph records side-crossing steps. The area growth $\operatorname{Area}B(r)=2\pi(\cosh r-1)$ and bounded tile diameter imply exponential growth of the number of tiles within $n$ steps: a unit geodesic segment meets a uniformly bounded number of tiles, by packing congruent interior balls, so a metric ball of radius proportional to $n$ lies in the step ball. Conversely the step ball lies in a metric ball of radius proportional to $n$. The regular right-angled pentagons with four around every vertex give the tiling $\{5,4\}$.
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