  We present new constrained and free-swimming experiments and simulations of a
pair of pitching hydrofoils interacting in a minimal school. The hydrofoils
have an out-of-phase synchronization and they are varied through in-line,
staggered, and side-by-side formations within the two-dimensional interaction
plane. It is discovered that there is a \textit{two-dimensionally} stable
equilibrium point for a side-by-side formation. In fact, this formation is
super-stable, meaning that hydrodynamic forces will passively maintain this
formation even under external perturbations and the school as a whole has no
net forces acting on it that cause it to drift to one side or the other.
Moreover, previously discovered \textit{one-dimensionally} stable equilibria
driven by wake vortex interactions are shown to be, in fact, two-dimensionally
\textit{unstable}, at least for an out-of-phase synchronization. Additionally,
it is discovered that a trailing-edge vortex mechanism provides the restorative
force to stabilize a side-by-side formation. The stable equilibrium is further
verified by experiments and simulations for freely-swimming foils where dynamic
recoil motions are present. When constrained, swimmers in compact side-by-side
formations experience collective efficiency and thrust increases up to 40\% and
100\%, respectively, whereas slightly staggered formations output an even
higher efficiency improvement of 84\% with a 87\% increase in thrust.
Freely-swimming foils in a stable side-by-side formation show an efficiency and
speed enhancement of up to 9\% and 15\%, respectively. These newfound schooling
performance and stability characteristics suggest that fluid-mediated
equilibria may play a role in the control strategies of schooling fish and
fish-inspired robots.
