  Consider a planar, bounded, $m$-connected region $\Omega$, and let
$\bord\Omega$ be its boundary. Let $\mathcal{T}$ be a cellular decomposition of
$\Omega\cup\bord\Omega$, where each 2-cell is either a triangle or a
quadrilateral. From these data and a conductance function we construct a
canonical pair $(S,f)$ where $S$ is a genus $(m-1)$ singular flat surface tiled
by rectangles and $f$ is an energy preserving mapping from ${\mathcal T}^{(1)}$
onto $S$.
