An overview of the theory and results on tessellations of three types of Riemann surfaces: the Euclidean plane, the sphere and the hyperbolic plane.
Roughly speaking, a tessellation of a space is a pattern which, repeated infinitely many times, fills the space without overlaps or gaps .
From a mathematical point of view, tessellations are pictorial representations of orbit spaces, corresponding to actions of groups on topological spaces. These can be of physical interest, for example the action on the hyperbolic plane, defined in §3, occurs in the theory of general relativity.
The expert provides an overview on tessellations for the hyperbolic plane.