along its canonical Dirichlet geodesic. Consequently, in addition to the foundational information provided, the construction also requires a description of the flute surfaces that are to be glued in. For each $i\in \Bbb{N},$ let $F_i$ be an untwisted flute. We shall henceforth assume that the infinite end on each of the untwisted flutes $F_i$ is of the 1st kind. One can provide sufficient conditions which guarantee this; for example, if the dividing loops grow sufficiently slowly, then the end is of the 1st kind.

Denote the sequence of dividing geodesics on $F_i$ by $\alpha_{i,k}$ where $k=0,1,2,\ldots$ We suppose that the length $|\alpha_{i,0}|$ of $\alpha_{i,0}$ is equal to the length $|I_i|$ of the interval $I_i$. On the flute $F_i$ orient the canonical Dirichlet geodesic $\beta_i^*$ in the direction of the infinite end of $F_i $ and so that $\beta_i^*(0)\in \alpha_{i,0}.$ Orient the geodesic ${\alpha }_{i,0}$ on $F_i$ so that the infinite end is to its right. Cut open $F_i$ along $\beta_i^*$ to get the complete hyperbolic surface $F_i^*$ with boundary. $F_i^*$ has the two boundary geodesics $\beta_i^1$ and $\beta_i^2$, where the cut open ${\alpha }_{i,0}$ is oriented from the the point $  \beta_i^1\cap{\alpha }_{i,0}$ to $  \beta_i^2\cap{\alpha }_{i,0}$. Each $\beta_i^j$ inherits 