Purcell's scallop theorem dictates that at low Reynolds number, only non-reciprocal strokes generate motion. Symmetry kills locomotion. A three-link swimmer — Purcell's original design — is the minimal system that breaks reciprocity through its two-hinged stroke cycle.
Micalizio, Morandotti, Shum, and Zoppello study what happens when this swimmer approaches a wall. Using resistive force theory with modified drag coefficients, they derive equations of motion that include the hydrodynamic interaction between the swimmer and a plane boundary. Geometric control theory then provides the analytical tool: controllability analysis at configurations parallel and tilted with respect to the wall.
For nearly-parallel configurations, the system is controllable — the swimmer can reach any nearby position and orientation. This extends the free-space result with the wall modifying drag coefficients but not breaking the geometric-control structure.
The surprise comes with tilted configurations. In free space, certain tilted orientations are controllability-degenerate: the swimmer's link geometry cannot generate the full range of motions. But near the wall, the hydrodynamic boundary condition introduces an asymmetry that rescues locomotion in precisely these degenerate cases. Tilted configurations that would forbid net displacement in unbounded fluid produce measurable displacement when a wall is present.
The wall does not just modify swimming efficiency. It qualitatively enables locomotion that the swimmer's internal geometry alone could not produce. The environmental asymmetry substitutes for the internal asymmetry the swimmer lacks, because the hydrodynamic boundary condition generates the geometric phase that the link configuration was missing.
A boundary can rescue locomotion that internal geometry forbids, because the wall introduces the symmetry-breaking that the swimmer's own stroke design fails to provide.