  A simple periodically driven system displaying rich behavior is introduced
and studied. The system self-organizes into a mosaic of static ordered regions
with three possible patterns, which are threaded by one-dimensional paths on
which a small number of mobile particles travel. These trajectories are
self-avoiding and non-intersecting, and their relationship to self-avoiding
random walks is explored. Near $\rho=0.5$ the distribution of path lengths
becomes power-law like up to some cutoff length, suggesting a possible critical
state.
