  Periodic solutions of the planar $N$-body problem determine braids through
the trajectory of $N$ bodies. Braid types can be used to classify periodic
solutions. According to the Nielsen-Thurston classification of surface
automorphisms, braids fall into three types: periodic, reducible and
pseudo-Anosov. To a braid of pseudo-Anosov type, there is an associated stretch
factor greater than 1, and this is a conjugacy invariant of braids. In 2006,
the third author discovered a family of multiple choreographic solutions of the
planar $2n$-body problem. We prove that braids obtained from the solutions in
the family are of pseudo-Anosov type, and their stretch factors are expressed
in metallic ratios. New numerical periodic solutions of the planar $2n$-body
problem are also provided.
