There are no generalized solutions to the three-body problem. Say you have three planets orbiting each other: it’s impossible to calculate how they will move using Newton’s laws of motion and gravity.

Unlike the two-body problem, the three-body problem has no general closed-form analytic solution. The differential equations that govern the motions of three gravitating bodies are not integrable and cannot be solved to give explicit formulas for the positions of the bodies as a function of time. For most initial conditions, the dynamical system for three orbiting bodies is chaotic, and the only way to predict their motions is to estimate them using numerical methods.

But thousands of stable three-body orbits have been found — their paths repeat after a given interval. The Three Body Orbits website is an atlas of some of those orbits: you can watch each three-body system play itself out in a wide variety of different shapes and complexities. There’s this simple figure 8, this slightly less simple one, and this one that looks like a spirograph.

Underlying Source: https://www.threebodyorbits.com/