Consider finitely many copies of a Euclidean rectangle and identify

edges of these rectangles pairwise, using translations or half-turns.

The result is a Riemann surface S equiped with a locally flat metric

for which the tessellation formed by the rectangles is naturally geodesic.

One can also define on S other metrics, in particular (possibly singular)

hyperbolic metrics for which the above tessellation is also geodesic in

the sense that the edges of the rectangles are geodesic arcs for these

metrics.

In many cases the tiling by rectangles allows to recover an equation for

the corresponding algebraic curve, providing a bridge between the algebraic

equation and the hyperbolic structure deduced from the multigeodesic

tessellation; in other words solving the uniformization problem for such

curves.