  We study the growing patterns in the rotor-router model formed by adding $N$
walkers at the center of a $L \times L$ two-dimensional square lattice,
starting with a periodic background of arrows, and relaxing to a stable
configuration. The pattern is made of large number of triangular regions, where
in each region all arrows point in the same direction. The square
circumscribing the region, where all the arrows have been rotated atleast one
full circle, may be considered as made up of smaller squares of different
sizes, all of which grow linearly with $N$, for $ 1 \ll N < 2 L$. We use the
Brooks-Smith-Stone-Tutte theorem relating tilings of squares by smaller squares
to resistor networks, to determine the exact relative sizes of the different
elements of the asymptotic pattern. We also determine the scaling limit of the
function describing the variation of number of visits to a site with its
position in the pattern. We also present evidence that deviations of the sizes
of different small squares from the linear growth for finite $N$ are bounded
and quasiperiodic functions of $N$.
