an idea of Siegel, and can be found in , and . Carathéodory gave an elementary treatment of the existence of tessellations by Schwarz triangles, i.e. tilings by geodesic triangles with angles /a, /b, /c with sum less than where a, b, c are integers. When all the angles equal /2g, this establishes the tiling by regular 4g-sided hyperbolic polygons and hence the existence of a particular compact Riemann surface of genus g as a quotient space. This special example, which has a cyclic group Z2g of bihomolomorphic symmetries, is used in the development below. The classification up to homeomorphism and diffeomorphism of compact Riemann surfaces implies the classification of closed orientable 2-manifolds up to homeomorphism and diffeomorphism: any two 2-manifolds with the same genus are diffeomorphic. In fact using a partition of unity, every closed orientable 2-manifold admits a Riemannian metric. For a compact Riemann surface a conformal metric can also be introduced which is conformal, so that in holomorphic coordinates the metric takes the form ρ(z) 2. Once this metric has been chosen, locally biholomorphic mappings are precisely orientation-preserving diffeomorphisms that are conformal, i.e. scale the metric by a smooth function. The existence of isothermal coordinates—which can be proved using either local existence theorems for the Laplacian or the Beltrami equation—shows that every closed oriented Riemannian 2-manifold can be given a complex structure compatible with its metric, and hence has the structure of a compact Riemann surface. This construction shows that the classification of closed orientable 2-manifolds up to diffeomorphism or homeomorphism can be reduced to the case of compact Riemann surfaces. The classification up to homeomorphism and diffeomorphism of compact Riemann surfaces can be accomplished using the fundamental polygon. Indeed, as Poincaré observed, convex fundamental polygons for compact Riemann surfaces H/Γ can be constructed by adapting the method of Dirichlet from the Euclidean space to hyperbolic space. Then following Nevanlinna and Jost, the fundamental domain can be modified in steps to yield a non-convex polygon with vertices lying in a single orbit of Γ and piecewise geodesic sides. The pairing relation on the sides is also modified in each of these steps. Each step involves cutting the polygon by a diagonal geodesic segment in the interior of the polygon and reassembling the polygon using one of the Möbius transformations involved in the pairing. No two paired sides can have a common vertex in the final pairing relation,