  Recent experiments and numerical simulations have shown that certain types of
microorganisms "reflect" off of a flat surface at a critical angle of
departure, independent of the angle of incidence. The nature of the reflection
may be active (cell and flagellar contact with the surface) or passive
(hydrodynamic) interactions. We explore the billiard-like motion of a body with
this empirical reflection law inside a regular polygon and show that the
dynamics can settle on a stable periodic orbit or can be chaotic, depending on
the swimmer's departure angle and the domain geometry. The dynamics are often
found to be robust to the introduction of weak random fluctuations. The
Lyapunov exponent of swimmer trajectories can be positive or negative, can have
extremal values, and can have discontinuities depending on the degree of the
polygon. A passive sorting device is proposed that traps swimmers of different
departure angles into separate bins. We also study the external problem of a
microorganism swimming in a patterned environment of square obstacles, where
the departure angle dictates the possibility of trapping or diffusive
trajectories.
