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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 special type of a (possibly immersed) genus $(m-1)$ singular flat surface, tiled by rectangles and $f$ is an energy preserving mapping from ${\\mathcal T}^{(1)}$ onto $S$. 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