Following Your Own Path: Designing Shapes That Can Roll Along Any Trajectory

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Imagine laying a cylinder on its side at the top of a ramp and letting it go. What do you imagine would happen? Nothing particularly exciting: it would roll in a straight line down the ramp. Now imagine if it was something more complicated—a cone, perhaps, or some elaborate polyhedron. Suddenly, the path begins to seem less predictable.

Mathematicians have long been investigating the rolling patterns of complicated solids such as these. But there is an even broader question: if you draw any kind of seemingly random periodic path, can you find a shape that would follow that trajectory when rolling down an incline? Recently, a team of mathematicians including Yaroslav Sobolev sought to explore this. “For me, it was just a useless but addictive puzzle that is easy to state, easy to think about, easy to solve, but which seems bottomless when you dig further into the underlying mathematics,” Sobolev said. 

He had first heard about shapes like sphericons and oloids from a Youtube video in 2020 and decided to create his own shapes. He first designed a trajectoid—a solid shape that periodically reverts to its original orientation when rolling down a slope—and then designed an algorithm that could compute the correct shape to match a given trajectory without having to search mathematical literature for solutions. “[The algorithm] succeeded as a joke and was promptly forgotten for the following 1.5 years,” Sobolev said.

However, the team recently published their work in Nature, claiming that for almost any given periodic path on an inclined plane, they could develop a trajectoid to roll along it. They accomplished this by developing an algorithm to determine the precise indentations needed on the solid to alter its path to fit the trajectory. They 3D-printed the shapes, embedded a steel ball for weight, and then experimentally validated their trajectoids by tracking their rolling paths and comparing them to the originally drawn ones (Figure 1).

But is this possible for any periodic path? The team found that for a given periodic path, a matching trajectoid would only exist if there exists a sphere which, after rolling along some number of the path’s periods, would restore its original orientation. “If your path satisfies this mathematical condition, computing the trajectoid’s shape [and manufacturing it] is pretty straightforward,” Sobolev said.

Surprisingly, they found that almost any random path fulfilled their condition, as long as the sphere was required to restore its original orientation after two periods. “For several months, we were trying to find even a single path for which this condition failed,” Sobolev said. The only paths that failed had a very specific symmetry and a sharp corner. Any path lacking these two characteristics satisfied the condition, and a matching trajectoid could be found.

As noted in the article, the existence of trajectoids for most paths could have unexpected implications in fields such as quantum and classical optics, which remain to be explored. There are also applications outside of science. Sobolov mentions the work of Henry Segerman, who created shapes that never roll in one direction and were used for circus performances. However, Sobolov notes that their study was largely motivated by a fundamental curiosity about shapes, their trajectories, and the unexpected things they can do.