In an aquarium the size of a dining table, two schools of black neon tetras swim in adjacent compartments separated by a transparent wall. They cannot touch each other, but they see every movement. And here's the surprising part: when one school decides to turn left, the other does the same a moment later. It's like two pendulums hanging behind glass, starting to swing in rhythm.

Fish in a school constantly change direction, but not chaotically. Their collective motion gets stuck in one of two stable states: either they all swim along the long side of the compartment in one direction, or in the opposite direction. It's like a swing that can rock left or right but doesn't linger in the middle. Switches between states happen spontaneously, driven by the random movements of individual fish. When the schools see each other, these switches begin to coincide.

The scientists measured how the strength of this synchronization depends on the width of the gap between compartments. It turned out that the coupling decays with distance according to a law that is slower than a typical exponential: the characteristic length was about 5 cm. This means that fish respond to a distant group not as individuals but as an averaged motion of the whole school. At short distances, details matter; at larger distances, the overall rhythm prevails.

The result is described by a model of two coupled bistable stochastic oscillators. In this model, each school is an oscillator with two stable states, and visual contact is the coupling that tends to align their phases. The model accurately reproduces the experimental data and predicts the mutual information between schools. This shows that living nature can be described by the same mathematical laws as inanimate oscillatory systems.

This work fills a gap in our understanding of collective behavior: previous models only considered local interactions between neighbors, whereas here, for the first time, the distance between groups is quantitatively linked to the phase synchronization of their collective orientations. This could help in developing systems for monitoring wild populations and in creating groups of robots that coordinate through optical sensors without physical contact. Two pendulums behind glass is not just a metaphor but a precise mathematical model of the living world.