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 t 0 and t 1 . That means its shape at the start and end are the same, i.e. r S ( t 0 ) = r S ( t 1 ) . Next, we consider motion obtained by time-reversal symmetry of the first motion that occurs during the period starting and ending at times t 2 and t 3 . using a similar mapping as in the previous section, we define t 2 = f ( t 1 ) and t 3 = f ( t 0 ) and define the shape in the reverse motion to be the same as the shape in the forward motion, r S ( t ) = r ′ S ( t ′ ) . Now we find the relationship between the net displacements in these two cases: