plane. Since $F$ has a single infinite end, the proof is complete. $ \hfill\Box $\

Gluing untwisted flutes  {#quilts}
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Flute surfaces on which the asymptotic geometry displays more diverse behavior than that exhibited by untwisted flutes can be constructed by gluing together untwisted flutes of the first kind that have been sliced open along their canonical Dirichlet geodesics. We shall describe a way to perform the construction to allow for infinitely many gluings along a superstructure of scaffolding curves defined by choosing a closed subset of an oriented circle of a given circumfrence.

The finite steps {#finite}
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Let $A$ denote the hyperbolic cylinder with the oriented simple closed geodesic $\alpha_0$ dividing $A$ into subsets $A^+$ and $A^-,$ where $A^+$ is to the right of $\alpha_0.$ $A$ is completely determined by the length of $\alpha_0,$ which we denote by $a=|\alpha_0|.$ Let $C$ be a closed subset of $\alpha_0$ and let $p$ be a distinguished point on $\alpha_0$. The complement of $C$ in $\alpha_0$ is a countable union of open geodesic segments which we refer to as intervals. Order the intervals lexicographically in terms of length (larger lengths precede smaller lengths) and oriented distance from $p,$ to get a sequence 