  Kumjian and Pask introduced an aperiodicity condition for higher rank graphs.
We present a detailed analysis of when this occurs in certain rank 2 graphs.
When the algebra is aperiodic, we give another proof of the simplicity of
$\ca(\Fth)$. The periodic C*-algebras are characterized, and it is shown that
$\ca(\Fth) \simeq \rC(\bT) \otimes \fA$ where $\fA$ is a simple C*-algebra.
