Hypocycloid Juggling Patterns

Craig S. Kaplan
Proceedings of Bridges 2016: Mathematics, Music, Art, Architecture, Education, Culture (2016)
Pages 71–78


When points are distributed evenly on a hypocycloid path and animated along that path, they can be seen as clustering together into “wheels”, groups of points that lie at the vertices of regular polygons and that rotate in synchrony. In some cases the points can group into two different sets of wheels, rotating in opposite directions. I derive formulas that predict the number of such wheels and the number of points on each one. The resulting animations are visually compelling and reminiscent of the motion of balls or clubs in multi-person juggling patterns.