In mathematics a translation surface is a surface obtained from identifying the sides of a polygon in the Euclidean plane by translations. An equivalent definition is a Riemann surface together with a holomorphic 1-form. These surfaces arise in dynamical systems where they can be used to model billiards, and in Teichmüller theory. A particularly interesting subclass is that of Veech surfaces (named after William A. Veech) which are the most symmetric ones. Definitions Geometric definition A translation surface is the space obtained by identifying pairwise by translations the sides of a collection of plane polygons. Here is a more formal definition. Let be a collection of (not necessarily convex) polygons in the Euclidean plane and suppose that for every side of any there is a side of some with and for some nonzero vector (and so that . Consider the space obtained by identifying all with their corresponding through the map . The canonical way to construct such a surface is as follows: start with vectors and a permutation on , and form the broken lines and starting at an arbitrarily chosen point. In the case where these two lines form a polygon (i.e. they do not intersect outside of their endpoints) there is a natural side-pairing. The quotient space is a closed surface. It has a flat metric outside the set images of the vertices. At a point in the sum of the angles of the polygons around the vertices which map to it is a positive multiple of , and the metric is singular unless the angle is exactly . Analytic definition Let be a translation surface as defined above and the set of singular points. Identifying the Euclidean plane with the complex plane one gets coordinates charts on with values in . Moreover, the changes of charts are holomorphic maps, more precisely maps of the form for some . This gives the structure of a Riemann surface, which extends to the entire surface by Riemann's theorem on removable singularities. In addition, the differential where is any chart defined above, does not depend on the chart. Thus these differentials defined on chart domains glue together to give a well-defined holomorphic 1-form on . The vertices of the polygon where the cone angles are not equal to are zeroes of (a cone angle of corresponds to a zero of order ). In the other direction, given a pair where