The figure-eight orbit that Nell Tyson mentions, as well as other newly discovered solutions to the old three-body problem, are so elegant that a name has been coined for them: choreographies. The figure eight is only the simplest example; other orbits are wildly more complex, with shapes that look more like fluttering butterflies. And those three-body orbits have been quickly generalized to even more fascinating dances for four or more bodies. Animated examples can be viewed on the Web at http://www.ams.org/new-in-math/cover/orbits1.html.The actual applet for a 3-body choreography is here.
As a theoretical astrophysicist working in stellar dynamics, I am sobered by the fact that great mathematicians and physicists worked on the three-body problem over the past three centuries without having any idea that orbits of this kind were awaiting discovery. And who knows what else there is to be found. It is not only with telescopes that new astronomical objects can be discovered. With even a small personal computer, a lucky guess, and enough persistence, anybody is now in a position to find new solutions to age-old problems of a kind that were completely beyond what Newton and the Le's and La's of celestial mechanics (Leverrier, Legendre, Lagrange, and Laplace, to name a few) could handle.
The Complex dance of orbits below mirrors another of Prof Hall's posts. His site is well worth reading for anyone into Spacecraft, Motorcycles, or Politics. Note that the mission it depicts was launched in 1978. We could do far better, with less fuel expended, today.