  We examine the collective dynamics of disks moving through a square array of
obstacles under cyclic square wave driving. Below a critical density we find
that system organizes into a reversible state in which the disks return to the
same positions at the end of every drive cycle. Above this density, the
dynamics are irreversible and the disks do not return to the same positions
after each cycle. The critical density depends strongly on the angle $\theta$
between the driving direction and a symmetry axis of the obstacle array, with
the highest critical densities appearing at commensurate angles such as
$\theta=0^\circ$ and $\theta=45^\circ$ and the lowest critical densities
falling at $\theta = \arctan(0.618)$, the inverse of the golden ratio, where
the flow is the most frustrated. As the density increases, the number of cycles
required to reach a reversible state grows as a power law with an exponent near
$\nu=1.36$, similar to what is found in periodically driven colloidal and
superconducting vortex systems.
