  A method is presented for constructing closed surfaces out of Euclidean
polygons with infinitely many segment identifications along the boundary. The
metric on the quotient is identified. A sufficient condition is presented which
guarantees that the Euclidean structure on the polygons induces a unique
conformal structure on the quotient surface, making it into a closed Riemann
surface. In this case, a modulus of continuity for uniformizing coordinates is
found which depends only on the geometry of the polygons and on the
identifications. An application is presented in which a uniform modulus of
continuity is obtained for a family of pseudo-Anosov homeomorphisms, making it
possible to prove that they converge to a Teichm\"uller mapping on the Riemann
sphere.
