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

Braids, metallic ratios and periodic solutions of the $2n$-body problem