  We give a combinatorial description of a family of 2-graphs which subsumes
those described by Pask, Raeburn and Weaver. Each 2-graph $\Lambda$ we consider
has an associated $C^*$-algebra, denoted $C^*(\Lambda)$, which is simple and
purely infinite when $\Lambda$ is aperiodic. We give new, straightforward
conditions which ensure that $\Lambda$ is aperiodic. These conditions are
highly tractable as we only need to consider the finite set of vertices of
$\Lambda$ in order to identify aperiodicity. In addition, the path space of
each 2-graph can be realised as a two-dimensional dynamical system which we
show must have zero entropy.
