  The simplest solutions of the N-body problem --symmetric relative
equilibria-- are shown to be organizing centers from which stem some recently
studied classes of periodic solutions. We focus on the relative equilibrium of
the equal-mass regular N-gon, assumed horizontal, and study the families of
Lyapunov quasi-periodic solutions bifurcating from them in the vertical
direction. The proof of the local existence of such solutions relies on the
fact that the restriction to the corresponding directions of the quadratic part
of the energy is positive definite. We then discuss the possibility of
continuing the families globally as action minimizers under symmetry
constraints by using the fact that, in rotating frames where they become
periodic, these solutions are highly symmetric. The paradigmatic examples are
the "Eight" families for an odd number of bodies and the "Hip-Hop" families for
an even number. We argue that it is precisely for these two families that
global minimization may be used. We also study the relation with the regular
N-gon, of the so-called "chain" choreographies (see C. Sim\'o, New families of
Solutions in N-Body Problems, Progr. Math. 201, 2001): here, only a local
minimization property is true (except for N=3) and moreover the parity plays a
deciding role, in particular through the value of the angular momentum.
