  A classic result of Brooks, Smith, Stone and Tutte \cite{BSST40} associates
to any finite planar network with distinguished source and sink vertices, a
tiling of a rectangle by smaller subrectangles whose aspect ratios are given by
the conductances of corresponding edges in the network. This tiling can be
viewed as a discrete analogue of the uniformizing conformal map that maps a
simply connected domain with four distinguished prime ends to a rectangle, so
that the four prime ends are mapped to the four corners of the rectangle. We
make this intuition precise by showing that if $\Omega$ is a simply connected
domain with four distinguished prime ends $A,B,C,D$ in counterclockwise order
and $(\Omega_{n})_{n\geq{1}}$ is a sequence of orthodiagonal maps with
distinguished boundary vertices $A_{n}, B_{n}, C_{n}, D_{n}$ in
counterclockwise order, that are finer and finer approximations of $\Omega$
with its distinguished boundary points $A,B,C,D$, then the corresponding
"rectangle tiling maps" converge uniformly on compacts to the aforementioned
conformal map on $\Omega$.
