  We prove the existence of a number of smooth periodic motions $u_*$ of the
classical Newtonian $N$-body problem which, up to a relabeling of the $N$
particles, are invariant under the rotation group ${\cal R}$ of one of the five
Platonic polyhedra. The number $N$ coincides with the order of ${\cal R}$ and
the particles have all the same mass. Our approach is variational and $u_*$ is
a minimizer of the Lagrangean action ${\cal A}$ on a suitable subset ${\cal K}$
of the $H^1$ $T$-periodic maps $u:{\bf R}\to {\bf R}^{3N}$. The set ${\cal K}$
is a cone and is determined by imposing to $u$ both topological and symmetry
constraints which are defined in terms of the rotation group ${\cal R}$. There
exist infinitely many such cones ${\cal K}$, all with the property that ${\cal
A}|_{\cal K}$ is coercive. For a certain number of them, using level estimates
and local deformations, we show that minimizers are free of collisions and
therefore classical solutions of the $N$-body problem with a rich
geometric-kinematic structure.
