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arXiv · 2406.16336

Tumbling Downhill along a Given Curve

Jean-Pierre Eckmann·Yaroslav I. Sobolev·Tsvi Tlusty
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Abstract

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.

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Jean-Pierre Eckmann, Yaroslav I. Sobolev, Tsvi Tlusty. 2024-06-24. Tumbling Downhill along a Given Curve. https://doi.org/10.1090/noti2947

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