Dancing polygons, rolling balls and the Cartan-Engel distribution

Abstract

A pair of planar polygons is "dancing" if one is inscribed in the other and they satisfy a certain cross-ratio relation at each vertex of the circumscribing polygon. Non-degenerate dancing pairs of closed $n$-gons exist for all $n\ge 6$. Dancing pairs correspond to trajectories of a non-holonomic mechanical system, consisting of a ball rolling, without slipping and twisting, along a polygon drawn on the surface of a ball 3 times larger than the rolling ball. The correspondence stems from reformulating both systems as piecewise rigid curves of a certain remarkable rank 2 non-integrable distribution defined on a 5-dimensional quadric in ${\mathbb{RP}}^6$, introduced by É. Cartan and F. Engel in 1893 in order to define the simple Lie group ${\mathrm{G}}_2$.

36 pages, 16 figures. Small corrections, remark 2 expanded. To appear in NY J of Math