Two bodies under Newtonian gravity can be beautifully simple: a planet and the Sun trace an ellipse. Add just one more body, and the mathematics changes dramatically. The three-body problem has no comparable tidy formula for the general motion. Its dynamics can become chaotic, so tiny differences in initial conditions may eventually lead to very different trajectories. This difficulty helped Poincaré develop a new way to study differential equations through the geometry and topology of their solutions, ideas that became foundational to modern nonlinear dynamics.