0 0.25 0.5 0.75 1 ! 0.4 ! 0.3 ! 0.2 ! 0.1 0 0.1 0.2 0.3 0.4 t ( a ) ( b ) Limit shape stomatocyte Stomatocyte-Oblate Self-intersecting stomatocyte v c$_{0}$ 2 1 0 $_{−}$1 $_{−}$2 0 ▷ 2 0 ▷ 3 0 ▷ 4 0 ▷ 5 0 ▷ 6 0 ▷ 7 U 〈 U$_{L}$ 〉 = − 0 ▷ 008 〈 U 〉 = − 0 ▷ 005 〈 U 〉 = − 0 ▷ 002 L FIG. 4: (color online) (a) Phase diagram for the spontaneous curvature model in the ( v, c 0 ) parameter space. Solid lines are our numerically-calculated lines that denote morphological transitions, while the dashed lines are qualitative, and adapted from Ref. [43]. (b) Three velocity profiles, corresponding to the elliptic paths through parameter space indicated in (a), with the largest velocities achieved along the elliptic path enclosing the greatest area. negative curvature. The increasing volume expels fluid from the cavity, and leads to a reversing of the sign of the vorticity. The overall sequence of asymmetric shapes is not time-reversible, leading to a net swimming velocity taking place in the − ˆ z direction. A phase diagram for the minimal energy shapes using the spontaneous curvature model is presented in Fig. 4a. The limit lines correspond to discontinuous morphological transitions, and therefore cannot be crossed in our quasi-static shape-change approach. One critical line corre- sponds to vesicles whose north and south poles self-intersect, and a second line corresponds to stomatocyte shapes that have a vanishing opening between the external fluid and the cavity within ( i.e. the shapes are two spheres, one contained entirely within the other). A third line marks the discontinuous phase transition between stomatocyte and oblate shapes. More details may be found in Ref. [43]. Beyond the symmetry constraints imposed by the Scallop theorem, other symmetry breaking is necessary in order for a body to achieve a net motion from a periodic shape cycle. Namely, the body surface must express fore-aft asymmetry in order to swim preferentially in any direction. Hence, parameter paths in the regions of phase space corresponding to prolate or oblate vesicle shapes cannot yield a net motion. However, paths which correspond to stomatocyte or pear shapes are fore/aft asymmetric and can swim. Since the area in phase space that contains pear shapes is very small, we will only examine the swimming stomatocytes. The largest elliptic path shown 14