especially, for smaller values of $m$ (see Fig. \[fig:animals\]). This set is not complete because discretization of a rod cannot produce polyominoes of all possible shapes but only the so-called linear polyominoes, i.e., those polyominoes possessing the property that a line can be drawn that intersects the interior of every square in the polyomino [@Friedman2001].

![Transformation of the continuous problem of rods into a “zoo of lattice animals” (linear polyominoes).\[fig:animals\]](supportinglatticeintersec){width="0.9\linewidth"}

Each occupied cell (a face of the lattice) was treated as conducting. To transform the mesh into a random resistor network (RRN), each cell was associated with a set of 4 conductors (Fig. \[fig:transformation\]). The electrical conductivity of the particles $\sigma_p$ was supposed to be much larger than the electrical conductivity of the substrate $\sigma_m$, i.e., the electrical conductivity contrast $\Delta=\sigma_p/\sigma_m \gg 1$. We put $\sigma_m =1$, and $\sigma_p = 10^6$ in arbitrary units. Different electrical conductivities corresponding to the empty cells, $\sigma_m$, occupied cells, $\sigma_p$, and between empty and occupied cells, $\sigma_{pm}=2\sigma_p \sigma_m / (\sigma_p+\sigma_m)$ were assumed (Fig. \[fig:transformation\]). In our calculations, the torus was unrolled, the two conducting buses were applied to the opposite borders of the lattice, and the electrical conductivity was calculated between these buses in 