The fact that the non-reciprocal shape change sequence N above produces a net change in the position of the car depends on the fact that the two transformations matrices involved do not commute. In particular, a pure translation does not commute with a pure rotation as can be seen from the following calculation:

This implies that if the car is driven in its forward direction by an amount e and then rotated about its center by the angle θ, the result will be different than had the car first been turned about its center by the angle θ and then driven forward by an amount e. The two final results are shown in the next plot with e = 1 and θ = 45 degrees.

This lack of commutation of the individual motions is exploited in the non-reciprocal shape change maneuver to ensure that the car moves as a result of each shape change cycle.