From Notices of the AMS
Tumbling Downhill Along a Given Curve
by
Jean-Pierre Eckmann
Yaroslov I. Sobolev
Tsvi Tlusty
Communicated by Chikako Mese
1. The Problem
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 ℝ2 can one carve a shape that will roll downhill following precisely this prescribed curve and its translationally repeated copies? See Fig. 1 for an example.
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.
2. Rolling Stones
There is an ample mathematical literature on rolling objects, but for our purpose, it may suffice to mention the study of rolling acrobatic apparatus [Seg21] and the rolling of balls on balls or Riemann surfaces [Lev93].
- Also in Notices
- Isometric Inversions and Applications
- Classical Values of Zeta, As Simple as Possible but Not Simpler
