 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 $\mathrm{Area}B(r)=2\pi (\cosh r-1)$ 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 $\{5,4\}$.

Tessellation is a geometric concept that refers to the covering of a plane with one or more geometric shapes, called tiles, without any overlaps or gaps. These shapes can be regular polygons, irregular shapes, or even complex figures. The key characteristics of a tessellation are that it must fill the entire surface without leaving any spaces between the tiles and the tiles may be rotated and flipped as long as they fit together seamlessly.