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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 aquadrilateral. From these data and a conductance function we construct acanonical pair $(S,f)$ where $S$ is a genus $(m-1)$ singular flat surface tiledby rectangles and $f$ is an energy preserving mapping from ${\mathcal T}^{(1)}$onto $S$.

Boundary Value Problems on Planar Graphs and Flat Surfaces with integer cone singularities, I: The Dirichlet Problem