  We study, theoretically and experimentally, a 1-parameter family of
transformations and their limiting vector field on the space of plane polygons.
These transformations are discrete analogs of completely integrable
transformation on closed plane curves, known as the bicycle correspondence,
that is a geometric realization of the B\"acklund transformation of the planar
filament equation. For odd-gons, we construct a symplectic form on the quotient
space by parallel translations and show that the transformations are
symplectic, and the vector field is Hamiltonian. In the case of triangles, we
prove complete integrability of the respective vector field and provide
evidence for the conjecture that the transformations are integrable as well.
