Asian Scientist Magazine (Sept. 21, 2023) —Imagine a ball rolling down a slope. If it doesn’t skid, it charts a straight path. Now imagine the trajectory of a 12-sided die rolling down. Clearly, that’s a much harder task. Mathematicians have been interested in figuring out the trajectories of complicated shapes such as the sphericon, a structure made by putting two cones together with a twist.
In a study published in Nature, a team of mathematicians and physicists from the Institute for Basic Science in South Korea flipped the challenge on its head. Given a random trajectory, is it possible to predict the shape of an object that would trace it down a slope?
They found that such shapes exist and dubbed them trajectoids. “We can design any shape that can roll based on the path we want them to run”, said Ruoyo Dong, one of the lead authors of the study.
For a trajectoid to exist, there are a few rules about what it should look like. For one, the center of gravity cannot move around a lot during the roll, necessitating a heavy center and little bumps and depressions on the trajectoid’s surface. The slope couldn’t be too steep, as slipping dominates rolling motion at greater inclines.
Finally, the path needs to be periodic. As the trajectoid traces this periodic path, it returns to its original orientation to ensure that the motion sustainably repeats itself. Consequently, if you connect all the points on the trajectoid that make contact with the ground, you get a closed loop on its surface.
The team observed that trajectoids that returned to the original orientation in a single run of their paths were rare. By making tiny modifications to the paths, however, the team could find trajectoids that worked for them.
“This shape can be found if the path that repeats has two or more periods except for very few exceptions,” said Dong.
With these constraints in place, the researchers developed an algorithm to design trajectoids that trace predefined paths. The logic is straightforward. Start with a sphere and split the path into small fragments. At every fragment, if required, mold the shape at the contact point slightly so as to make it move to the next fragment. Repeat this process across all fragments and you end up with a trajectoid that traces the desired path.
To experimentally validate the algorithm, the team 3D-printed the trajectoids. They inserted heavy ball bearings inside these shapes to ensure a fixed center of gravity. The shapes followed the predefined paths, even when trajectories went uphill over short stretches.
While driven purely by mathematical curiosity, this research may find use in other sciences. For example, in theoretical physics, electrons have a spin component that can be modeled with trajectoids. Changes in the orientation of a trajectoid along its path could serve as a proxy for the evolution of the angle of an electron over time.
A more practical application of trajectoids lies in soft robotics. Robots that trace simpler trajectories, such as those made by sphericons, exist already. This work could pave the way for the development of robots that trace more complicated paths. Think of shape-shifting robots that incorporate aspects of trajectoidal geometries.
Source: Institute for Basic Science, South Korea ; Image:
The paper can be found at: Solid-body trajectoids shaped to roll along desired pathways | Nature
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