  Since the discovery of the figure-8 orbit for the three-body problem [Moore
1993] a large number of periodic orbits of the n-body problem with equal masses
and beautiful symmetries have been discovered. However, most of those that have
appeared in the literature are either planar or are obtained from perturbations
of planar orbits. Here we exhibit a number of new three-dimensional periodic
n-body orbits with equal masses and cubic symmetry. We found these orbits
numerically by minimizing the action as a function of the trajectories' Fourier
coefficients. We also give numerical evidence that a planar 3-body orbit first
found in [H\'enon, 1976], rediscovered by [Moore 1993], and found to exist for
different masses by [Nauenberg 2001], is dynamically stable.
  It is a pleasure to dedicate this paper to Philip Holmes.
