Beyond planar: fish schools adopt ladder formations in 3D

The coordinated movement of fish schools has long captivated researchers studying animal collective behavior. Classical literature from Weihs and Lighthill suggests that fish schools should favor planar diamond formations to increase hydrodynamic efficiency, inspiring a large body of work ranging from fluid simulations to hydrofoil experiments. However, whether fish schools actually adopt and maintain this idealized formation remains debated and unresolved. When fish schools are free to self-organize in three dimensions, what formations do they prefer? By tracking polarized schools of giant danios (Devario aequipinnatus) swimming continuously for ten hours, we demonstrate that fish rarely stay in a horizontal plane, and even more rarely, in the classical diamond formation. Of all fish pairs within four body-lengths from each other, only 25.2% are in the same plane. Of these, 54.6% are inline, 30.0% are staggered, and 15.4% are side-by-side. The diamond formation was observed in less than 0.1% of all frames. Notably, a "ladder formation" emerged as the most probable formation for schooling giant danios, appearing in 79% of all fish pairs and fish schools elongate at higher swimming speeds. These findings highlight the dynamic and three-dimensional nature of fish schools and suggest that hydrodynamic benefits may be obtained without requiring fixed positions. This research provides a foundation for future studies that examine the hydrodynamics and control of underwater collectives in 3D formations.