Closed geodesics on 2-dimensional x-geometric polyhedra

For 2-dimensional finite chi-geometric polyhedra of curvature K less than or equal to chi < 0 it is shown that the polygonal flow, applied to a closed curve, converges to a geodesic. Moreover, it is shown that there exists a finite number of closed geodesics with length smaller than a given positive B. As an application of the polygonal flow, a way of constructing closed, in particular simple, curves is given as well as a condition which implies that a curve is non-homotopic to a point.