  A well-known theorem of Rodin \& Sullivan, previously conjectured by
Thurston, states that the circle packing of the intersection of a lattice with
a simply connected planar domain $\Omega$ into the unit disc $\mathbb{D}$
converges to a Riemann map from $\Omega$ to $\mathbb{D}$ when the mesh size
converges to 0. We prove the analogous statement when circle packings are
replaced by the square tilings of Brooks et al.
