text: If a swimmer is moving in a periodic fashion that is time invariant, we know that the average displacement during one period must be zero. To illustrate the proof, let us consider a swimmer deforming during one period that starts and ends at times  and . That means its shape at the start and end are the same, i.e.  Next, we consider motion obtained by time-reversal
  symmetry of the first motion that occurs during the period starting and ending at times  and  using a similar mapping as in the previous section, we define  and  and define the shape in the reverse motion to be the same as the shape in the forward motion,  Now we find the relationship between the net displacements in these two cases:
  This is the second key result. Combining with our first key result from the previous section, we see that  We see that a swimmer that reverses its motion by reversing its sequence of shape changes leads to the opposite distance traveled. In addition, since the swimmer exhibits reciprocal body deformation, the sequence of motion is the same between  and  and  and  Thus, the distance traveled should
  be the same independently of the direction of time, meaning that reciprocal motion cannot be used for net motion in low Reynolds number environments.