 

Gil Bor and
Luis Hernández Lamoneda
Dancing polygons, rolling balls, and the CartanEngel distribution
view
print


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


Abstract
A pair of planar polygons is 'dancing' if one is inscribed in the other and they satisfy a certain crossratio relation at each vertex of the circumscribing polygon. Nondegenerate dancing pairs of closed ngons exist for all
n>= 6. Dancing pairs correspond to trajectories of a nonholonomic 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 nonintegrable distribution defined on a 5dimensional quadric in RP^{6}, introduced by É. Cartan and
F. Engel in 1893 in order to define the simple Lie group G_{2}.


Acknowledgements
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 A1S45886.
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
CIMAT
A.P. 402
Guanajuato, Gto. 36000, Mexico
gil@cimat.mx
Luis Hernández Lamoneda
CIMAT
A.P. 402
Guanajuato, Gto. 36000, Mexico
lamoneda@cimat.mx

