a parameterization from $\beta_i^*.$ By re-identifying the boundary geodesics $\beta_i^1$ and $\beta_i^2$ so that $\beta_i^1(t)$ is glued to $\beta_i^2(t),$ for $t\in {{\mathbb{R}}},$ the starting flute $F_i$ will be reproduced.

On $F^*_i$ we shall continue to refer to the cut open dividing loops $\alpha_{i,k}$ by the same names. Observe that the involution $r$ of $F_i$ induces an isometric involution $r^*$ of $F_i^*$, interchanging $\beta_i^1(t)$ and $\beta_i^2(t)$ for $t\in {{\mathbb{R}}}$, fixing the $\gamma$-curves and mapping each $\alpha_{i,k}$ onto itself.

With the data consisting of $A,\, p,\, C$ and the surfaces $F_i$, we define a sequence of flute surfaces $S_n,$ constructed by a succession of single replacements of the type described at the beginning of the section, as follows. Set $S_0=A.$ Suppose that the hyperbolic surface $S_i$ has been defined for some $i\geq0,$ and has the property that $S_{i}$ contains an isometric copy of the original scaffolding, so that the union of the scaffolding and the strips $E_k$ with $k>i$ embeds isometrically in $A$. (Here objects with the same name are identified by the embedding.)

In order to construct $S_{i+1}$ begin by removing the interior of the strip $E_{i+1}$ from $S_i$. Insert $F_{i+1}^*$ in place of $E_{i+1}$ by identifying the two pairs of 