Dirichlet polygon, a description will be given of the method of , elaborated in , for modifying the polygon to a non-convex polygon with 4g equivalent vertices and a canonical pairing on the sides. This treatment is an analytic counterpart of the classical topological classification of orientable 2-dimensional polyhedra presented in . Fricke canonical polygon Given a Riemann surface of genus g greater than one, Fricke described another fundamental polygon, the Fricke canonical polygon, which is a very special example of a Dirichlet polygon. The polygon is related to the standard presentation of the fundamental group of the surface. Fricke's original construction is complicated and described in . Using the theory of quasiconformal mappings of Ahlfors and Bers, gave a new, shorter and more precise version of Fricke's construction. The Fricke canonical polygon has the following properties: The vertices of the Fricke polygon has 4g vertices which all lie in an orbit of Γ. By vertex is meant the point where two sides meet. The sides are matched in distinct pairs, so that there is a unique element of Γ carrying a side to the paired side, reversing the orientation. Since the action of Γ is orientation-preserving, if one side is called , then the other of the pair can be marked with the opposite orientation . The edges of the standard polygon can be arranged so that the list of adjacent sides takes the form . That is, pairs of sides can be arranged so that they interleave in this way. The sides are geodesic arcs. Each of the interior angles of the Fricke polygon is strictly less than , so that the polygon is strictly convex, and the sum of these interior angles is 2. The above construction is sufficient to guarantee that each side of the polygon is a closed (non-trivial) loop in the Riemann surface H/Γ. As such, each side can thus an element of the fundamental group . In particular, the fundamental group has 2g generators , with exactly one defining constraint, . The genus of the Riemann surface H/Γ is g. Area The area of the standard fundamental polygon is where g is the genus of the Riemann surface (equivalently, where 4g is the number of the sides of the polygon). Since the standard polygon is a representative of H/Γ, the total area of the Riemann surface is equal to the area of the