  The present work studies the continuation class of the regular $n$-gon
solution of the $n$-body problem. For odd numbers of bodies between $n = 3$ and
$n = 15$ we apply one parameter numerical continuation algorithms to the
energy/frequency variable, and find that the figure eight choreography can be
reached starting from the regular $n$-gon. The continuation leaves the plane of
the $n$-gon, and passes through families of spatial choreographies with the
topology of torus knots. Numerical continuation out of the $n$-gon solution is
complicated by the fact that the kernel of the linearization there is high
dimensional. Our work exploits a symmetrized version of the problem which
admits dense sets of choreography solutions, and which can be written as a
delay differential equation in terms of one of the bodies. This symmetrized
setup simplifies the problem in several ways. On one hand, the direction of the
kernel is determined automatically by the symmetry. On the other hand, the set
of possible bifurcations is reduced and the $n$-gon continues to the eight
after a single symmetry breaking bifurcation. Based on the calculations
presented here we conjecture that the $n$-gon and the eight are in the same
continuation class for all odd numbers of bodies.
