Periodic orbits of the N-body problem with the symmetry of platonic polyhedra.

We review some recently discovered periodic orbits of the N-body problem, whose existence is proved by means of variational methods. These orbits are minimizers of the Lagrangian action functional in a set of $T$-periodic loops, equivariant for the action of a group $G$ and satisfying some topological constraints. Both the group action and the topological constraints are defined using the symmetry of Platonic polyhedra.