Researchers discover algorithm to create shapes that roll down pre-determined paths

Lopsided solids promise applications in quantum mechanics and medicine

Researchers have developed a method to construct solid objects that roll down pre-determined paths, which they reckon could have applications in quantum mechanics and medicine.

The team of researchers were led by Bartosz Grzybowski, distinguished professor at the Institute for Basic Science in Korea, and included first authors Yaroslav Sobolev and Ruoyu Dong.

Starting with the basics, a perfect cylinder rolls down an inclined plane in a straight line, whereas a cone rolls in a curve. To get a ball of malleable clay to roll down a simple path, you can force it down a specific path once, squashing it as you go. Take it to the top again, restart it from the initial starting point on the ball's surface, and it will roll down the same path.

The researchers took this principle to develop an algorithm that could produce a shape capable of following almost any pre-determined path, even making the weird-shaped solids out of 3D-printed plastic and solid ball-bearings (for weight) to prove the point.

The resulting objects, which they call "trajectoids," could have applications in understanding quantum mechanics and helping improve magnetic medical imaging.

"Our study is motivated largely by fundamental curiosity, but the existence of trajectoids for most paths has unexpected implications," their research paper, published in the journal Nature today, said.

To achieve the results, the researchers had to overcome one key sticking point. It was difficult to design a shape that followed a pre-determined path ending on the same point on the object's surface where it started before following the same path again.

To get around the problem, the researchers found that designing an object that rotated twice over a path could accommodate almost every path imaginable. These were called "two-period trajectoids."

In an accompanying article, Elisabetta Matsumoto, post-doctoral fellow at the School of Physics, Georgia Institute of Technology, and Henry Segerman, assistant professor at the Department of Mathematics, Oklahoma State University, said:

"This approach is surprising, and suggests that there should be a precise mathematical statement saying exactly when a two-period trajectoid exists. The authors provide an example curve that doesn’t work, but also show that tiny modifications to that curve make it work. They conjecture that paths that don't work are infinitely rare. It therefore seems likely that any designer wanting to use a trajectoid in a real-world application would not run into problems in constructing one. However, future work developing a more precise mathematical understanding of the issue would help to connect this work to applications, as well as to open up more purely mathematical veins of research."

The research could have applications in robotics and beyond.

In physics, for example, the angular moment, or spin, on an electron "can point in any direction, so the curved 'tabletop' trajectory of a trajectoid could represent the orientation of a spin as a function of time."

"In quantum computing, this representation could be used to control the evolution of a quantum bit – the basic unit of quantum information, which can be encoded in spin," they said.

A similar technique could help reduce unwanted noise in medical magnetic resonance imaging. "Trajectoids that have been engineered to control the spin dynamics – and therefore the magnetic fields – in these devices could be used to help separate useful signals from noise," the commentators said.

"Whether or not these applications materialize, Sobolev and colleagues' algorithm offers an insightful answer to the problem of how to encode an object's trajectory using its shape alone," they added. ®

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