a translation surface is defined geometrically by taking a collection of polygons in the Euclidean plane and identifying faces by maps of the form (a "half-translation"). Note that a face can be identified with itself. The geometric structure obtained in this way is a flat metric outside of a finite number of singular points with cone angles positive multiples of . As in the case of translation surfaces there is an analytic interpretation: a half-translation surface can be interpreted as a pair where is a Riemann surface and a quadratic differential on . To pass from the geometric picture to the analytic picture one simply takes the quadratic differential defined locally by (which is invariant under half-translations), and for the other direction one takes the Riemannian metric induced by , which is smooth and flat outside of the zeros of . Relation with Teichmüller geometry If is a Riemann surface then the vector space of quadratic differentials on is naturally identified with the tangent space to Teichmüller space at any point above . This can be proven by analytic means using the Bers embedding. Half-translation surfaces can be used to give a more geometric interpretation of this: if are two points in Teichmüller space then by Teichmüller's mapping theorem there exists two polygons whose faces can be identified by half-translations to give flat surfaces with underlying Riemann surfaces isomorphic to respectively, and an affine map of the plane sending to which has the smallest distortion among the quasiconformal mappings in its isotopy class, and which is isotopic to . Everything is determined uniquely up to scaling if we ask that be of the form , where , for some ; we denote by the Riemann surface obtained from the polygon . Now the path in Teichmüller space joins to , and differentiating it at gives a vector in the tangent space; since was arbitrary we obtain a bijection. In facts the paths used in this construction are Teichmüller geodesics. An interesting fact is that while the geodesic ray associated to a flat surface corresponds to a measured foliation, and thus the directions in tangent space are identified with the Thurston boundary, the Teichmüller geodesic ray associated to a flat surface does not always converge to the corresponding point on the boundary, though almost all such rays do so. Veech surfaces The Veech group If is a translation surface its Veech