  First, we apply Thurston's construction of pseudo-Anosov homeomorphisms to
grid graphs and obtain translation surfaces whose Veech groups are
commensurable to $(m,n,\infty)$ triangle groups. These surfaces were first
discovered by Bouw and M\"oller, however our treatment of the surfaces differs.
We construct these surfaces by gluing together polygons in two ways. We use
these elementary descriptions to compute the Veech groups, resolve primitivity
questions, and describe the surfaces algebraically. Second, we show that some
$(m,n, \infty)$ triangle groups can not arise as Veech groups. This generalizes
work of Hubert and Schmidt.
