boundary at infinity of a Dirichlet polygon for a quilted surface group. To this end we prove

\[C\] Let $K$ be a compact subset of ${{\mathbb{R}}}$. There is a Fuchsian group $G$ representing a quilted flute surface and a point $\tilde{p}\in{\Bbb{H}}^2$ so that the boundary at infinity of the Dirichlet polygon for $G$ centered at $\tilde{p}$ consists of the union of the set $K$ and a countable set of isolated parabolic fixed points of $G$.

The paper is structured as follows. We begin in Section \[review\], with a review of some facts about flute surfaces. In Section \[quilts\] we construct quilted flute surfaces and derive some of their basic properties. In Section \[number4\] we investigate a special class of canonical critical rays that go out the infinite end of a quilted flute surface. The complexity of the set of such rays is one measure of the complexity of the surface. Before moving on to probe more results about quilted surfaces we must turn to the hyperbolic plane. In Section \[lemmas\] we prove several lemmas, crucial to the proofs in later sections. In Section \[example2\] we develop an intrinsic version of Theorem \[C\] for quilted surfaces and show how this can 