New York Journal of Mathematics
Volume 29 (2023), 981-1015


Gil Bor and Luis Hernández Lamoneda

Dancing polygons, rolling balls, and the Cartan-Engel distribution

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Published: August 5, 2023.
Keywords: (2,3,5)-distribution; simple group G2; projective polygon pairs; rolling distribution.
Subject [2010]: 58A30;53A20; 53A40; 53A55.

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.

Author information

Gil Bor
A.P. 402
Guanajuato, Gto. 36000, Mexico


Luis Hernández Lamoneda
A.P. 402
Guanajuato, Gto. 36000, Mexico