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A method is presented for constructing closed surfaces out of Euclideanpolygons with infinitely many segment identifications along the boundary. Themetric on the quotient is identified. A sufficient condition is presented whichguarantees that the Euclidean structure on the polygons induces a uniqueconformal structure on the quotient surface, making it into a closed Riemannsurface. In this case, a modulus of continuity for uniformizing coordinates isfound which depends only on the geometry of the polygons and on theidentifications. An application is presented in which a uniform modulus ofcontinuity is obtained for a family of pseudo-Anosov homeomorphisms, making itpossible to prove that they converge to a Teichm\"uller mapping on the Riemannsphere.

Paper folding, Riemann surfaces, and convergence of pseudo-Anosov sequences