  Peach introduced rhombal algebras associated to quivers given by tilings of
the plane by rhombi. We develop general techniques to analyse rhombal algebras,
including a filtration by what we call rhombus modules.
  We introduce a way to relate the infinite-dimensional rhombal algebra
corresponding to a complete tiling of the plane to finite-dimensional algebras
corresponding to finite portions of the tiling. Throughout, we apply our
general techniques to the special case of the Rauzy tiling, which is built in
stages reflecting an underlying self-similarity. Exploiting this self-similar
structure allows us to uncover interesting features of the associated
finite-dimensional algebras, including some of the tree classes in the stable
Auslander-Reiten quiver.
