# Tumbling Downhill along a Given Curve

Jean-Pierre Eckmann, Yaroslav I. Sobolev, Tsvi Tlusty

Jean-Pierre Eckmann, Yaroslav I. Sobolev, Tsvi Tlusty

**24/6/24**Published in : arXiv:2406.16336

A cylinder will roll down an inclined plane in a straight line. A cone will roll around a circle on that plane and then will stop rolling. We ask the inverse question: For which curves drawn on the inclined plane \mathbb{R}^2 can one carve a shape that will roll downhill following precisely this prescribed curve and its translationally repeated copies? This simple question has a solution essentially always, but it turns out that for most curves, the shape will return to its initial orientation only after crossing a few copies of the curve - most often two copies will suffice, but some curves require an arbitrarily large number of copies.