  Higher-rank graphs (or $k$-graphs) were introduced by Kumjian and Pask to
provide combinatorial models for the higher-rank Cuntz-Krieger $C^*$-algebras
of Robertson and Steger. Here we consider a family of finite 2-graphs whose
path spaces are dynamical systems of algebraic origin, as studied by Schmidt
and others. We analyse the $C^*$-algebras of these 2-graphs, find criteria
under which they are simple and purely infinite, and compute their $K$-theory.
We find examples whose $C^*$-algebras satisfy the hypotheses of the
classification theorem of Kirchberg and Phillips, but are not isomorphic to the
$C^*$-algebras of ordinary directed graphs.
