  Steady swimming appears both periodic and stable. These characteristics are
the very definition of limit cycles, and so we ask "Can we view swimming as a
limit cycle?" In this paper we will not be able to answer this question in
full. However, we shall find that reduction by SE(d)-symmetry brings us closer.
Upon performing reduction by symmetry, we will find a stable fixed point which
corresponds to a motionless body in stagnant water. We will then speculate on
the existence of periodic orbits which are "approximately" limit cycles in the
reduced system. When we lift these periodic orbits from the reduced phase
space, we obtain dynamically robust relatively periodic orbits wherein each
period is related to the previous by an SE(d)-phase. Clearly, an SE(d) phase
consisting of nonzero translation and identity rotation means directional
swimming, while non-trivial rotations correspond to turning with a constant
turning radius.
