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>= 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 RP⁶, introduced by É. Cartan and F. Engel in 1893 in order to define the simple Lie group G₂.

We thank Robert Bryant for informative correspondence and to Travis Wilse for reading an initial draft and making useful suggestions. We acknowledge support from CONACYT Grant A1-S-45886. LHL thanks the Mathematics Department of the University of Santiago de Compostela for its hospitality while portions of this article were done.