  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 $n\geq 6$. 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 \'E. Cartan and F.
Engel in 1893 in order to define the simple Lie group $\mathrm{G}_2$.
