Each path has its own rolling stone

If you roll a tube of glue on an inclined surface, its path traces a straight line. The ice cream cone follows the curve of a semicircle. Other shapes have a more surprising trajectory, such as a spherical shape that adopts a sinusoidal oscillation. But, instead of starting with the object to determine its path, is it possible to design an object that will follow a predetermined path? That’s what Yaroslav Sobolev and his colleagues at the Institute of Basic Sciences in Ulsan, South Korea, did: they showed that on almost every imaginable path, there is an object rolling along that path. They even developed an algorithm to design these objects, called trajectories.

The idea came to Yaroslav Sobolev by manually shaping a ball of clay. By making the ball follow the path he wanted it to take and gently pressing it to flatten its surface, the researcher obtained an object that followed the path well when it rolled.

The algorithm proposed by the researchers is based on this measurement. But instead of sculpting the initial ball by rolling it, it is cut. For example, if the track consists of a straight line, then the ball in this part should behave like a cylinder and be cut accordingly. “Once you have all this shavings done, you have a complex but well-defined structure as a mathematical object. The remaining challenge is to print it in 3D,” notes Jean-Pierre Ekman, a researcher in mathematics and theoretical physics at the University of Geneva, who participated in the study. In 3D, you have to go through the triangulation step (cutting the surface into small triangles).

Here, the ball is made of plastic with a steel core. By using a very dense material in the centre, the researchers ensure that the center of mass does not shift much, regardless of what is removed from the surface. This ensures that the body rolls without stopping on a slightly inclined plane.

“So for almost any curve – even those with loops, with a part going downhill, or with corners, that is, sudden changes in direction – a path can be drawn,” the researcher announced. Only in some rare cases, such as fractal (infinitely cut) graphics, does the algorithm not work.

There is only one condition to make sure it works: you must draw the same path twice before giving it to the algorithm. If you trace the path only once, there is a risk that the start point and end point of the path on the path do not match. “But what we want is for our body to be able to roll indefinitely following this path and repeating it!”, explains Jean-Pierre Ekman. If the two points are not in the same place on the body, it may stop or follow another path, once the path is crossed. “But Experimentally, we have discovered that in most cases, it is enough to draw the same path twice for the path to reorient itself correctly after rolling,” the researcher explains. Why are two lines generally enough? And that’s what interests the researchers now. “It’s an open mathematical problem, and it’s interesting Extremely!”,” he admitted.

In the meantime, the researchers have made their code available (on the website https://github.com/yaroslavsobolev/trajectoids) so that you too can chart the course of your invention. To do this, you just need to get the coordinates of the path you want. So, get to your 3D printers!