Translation Surface

A translation surface is constructed by taking a collection of polygons in the Euclidean plane and gluing pairs of their sides together by translation, so that each identified pair is related by a vector that maps one side onto the other. Equivalently, a translation surface can be described as a Riemann surface equipped with a holomorphic 1-form, with the surface's singular points, where the cone angle is not the usual 2 pi, corresponding to the zeros of that differential form; the simplest example, gluing opposite sides of a parallelogram, produces a flat torus with no singular points at all. Translation surfaces carry a flat metric inherited from the plane and are used to study billiard flows, geodesic flows and Teichmuller theory, making them a central tool for understanding chaotic behavior in dynamical systems.