Recently, Alain Chenciner (Paris 7; Bureau des Longitudes) and the speaker discovered a surprisingly simple periodic orbit for the Newtonian three-body problem : three equal masses chase each other around a fixed curve in the plane with the shape of a figure eight. From many points of view this `eight' is the simplest periodic solution for the problem, after those of Lagrange and Euler. We outline our existence proof, which combines the direct method of the calculus of variations, the use of discrete symmetries and a detailed knowledge of the geometry of space of oriented congruence classes of planar triangles.

We also plan to describe the results of recent numerical investigations by Carles Simo (University of Barcelona, Spain). Simo has found that our orbit is KAM-stable. He has also found that it is the first member of an apparently infinite collection of N-body orbits, N arbitrary. An orbit in this family is a solution to Newton's equations (1/r potential) consisting of N equal masses chasing each other around a fixed planar curve $c_N$. The curves found have the shapes of eights, chains, flowers, spirograph patterns, and the number appearing for a given N grows exponentially in N. We hope to describe the beginnings of a theory to explain the existence of all of these new solutions.