is a compact Riemann surface and a holomorphic 1-form one can construct a polygon by using the complex numbers where are disjoint paths between the zeroes of which form an integral basis for the relative cohomology. Examples The simplest example of a translation surface is obtained by gluing the opposite sides of a parallelogram. It is a flat torus with no singularities. If is a regular -gon then the translation surface obtained by gluing opposite sides is of genus with a single singular point, with angle . If is obtained by putting side to side a collection of copies of the unit square then any translation surface obtained from is called a square-tiled surface. The map from the surface to the flat torus obtained by identifying all squares is a branched covering with branch points the singularities (the cone angle at a singularity is proportional to the degree of branching). Riemann–Roch and Gauss–Bonnet Suppose that the surface is a closed Riemann surface of genus and that is a nonzero holomorphic 1-form on , with zeroes of order . Then the Riemann–Roch theorem implies that If the translation surface is represented by a polygon then triangulating it and summing angles over all vertices allows to recover the formula above (using the relation between cone angles and order of zeroes), in the same manner as in the proof of the Gauss–Bonnet formula for hyperbolic surfaces or the proof of Euler's formula from Girard's theorem. Translation surfaces as foliated surfaces If is a translation surface there is a natural measured foliation on . If it is obtained from a polygon it is just the image of vertical lines, and the measure of an arc is just the euclidean length of the horizontal segment homotopic to the arc. The foliation is also obtained by the level lines of the imaginary part of a (local) primitive for and the measure is obtained by integrating the real part. Moduli spaces Strata Let be the set of translation surfaces of genus (where two such are considered the same if there exists a holomorphic diffeomorphism such that ). Let be the moduli space of Riemann surfaces of genus ; there is a natural map mapping a translation surface to the underlying Riemann surface. This turns into a locally trivial fiber bundle over the moduli space. To a compact translation surface there is associated the data where are the