Researchers in South Korea and Switzerland have developed an algorithm that creates 3D objects that follow specific meandering paths as they roll downhill. They have also shown that their technique could be used to develop new control protocols for seemingly unrelated systems including quantum spins and the polarization of light.
Rolling objects have played key roles in technology since at least the advent of the wheel. Most rolling objects used by humans are cylindrical, spherical or conical. The first two shapes are useful because they tend to roll in straight lines, whereas conical shapes are used when a circular trajectory is needed.
However, there are also objects that will roll downhill forever while following repeating, meandering paths – a simple example being a sinusoidal path. These objects include oloids, sphericons, polycons, platonicons and two-circle rollers. Some of these have been used in robotics and also for mixing materials. Beyond these practical applications, the discovery and characterization of shapes that take meandering paths is an interesting mathematical problem.
Seeking trajectoids
Now, Bartosz Grzybowski at the Institute for Basic Science in Ulsan and colleagues have sought to solve a mathematical problem that generalizes the search for such objects – which they have dubbed “trajectoids”. They have also successfully made some of these trajectoids using 3D printing.
Writing in the journal Nature, the team states the problem as “given an infinite periodic trajectory, find the shape that would trace this trajectory when rolling down a slope”.
The team showed that a potential trajectoid can be described by a virtual exercise that involves drawing a periodic trajectory on a flat surface. Then, a sphere is rolled over the surface such that the line is transferred to the surface of the sphere. If the start of the trajectory matches up with the end of the trajectory – thereby creating a continuous loop on the surface of the sphere – then it should be possible to create a trajectoid that follows that route. The team also found that when trajectories do not match up, they can be tweaked to do so.
Two or more periods
Although this technique can be used to identify suitable trajectoid paths, the researchers discovered that fitting one period of a trajectory onto a sphere was actually a difficult thing to do. In contrast, they found that it was much easier to fit two (or more) periods of a trajectory onto a sphere. Indeed, the team surmises that this technique should work for just about every possible repeating path – showing that the number of paths that cannot be mapped after two or more rotations are exceeding rare.
Once they had perfected their method to identify trajectoid paths, the researchers devised a scheme for fabricating the corresponding trajectoids. In their technique, an ideal trajectoid begins as a dense spherical core with a concentric outer shell that has zero density. The desired trajectory is split into a series of linear segments. To have the object roll along a linear segment, part of the outer shell is “shaved off” to create a small region that has cylindrical curvature and will therefore only roll along the direction of the line segment (assuming no slippage).
This process is repeated for all successive linear segments. This creates a trajectoid that is a combination of cylindrical surfaces, all of which have axes of rotation that are parallel to the rolling plane and go through the centre of mass of the object.
3D printed shells
The team then created such trajectoids using 3D printing to create low density outer shells. These were printed in hemispheres that were then glued onto heavy steel balls with much higher densities. The trajectoids were then rolled down an incline that was covered with sandpaper to prevent slippage.
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The team tested a number of different trajectoids and found that many of them did a very good job at following their expected downhill routes. Others, however, came to a halt, while some trajectoids struggled to negotiate sharp turns in their predicted paths.
The process of translating a repeating meandering trajectory onto a sphere is similar to how the evolution of some quantum systems is described in terms of the trajectory of a point on a “Bloch sphere”. Examples of this include the description of how a nuclear spin is manipulated in a nuclear magnetic resonance (NMR) measurement, or how an electronic spin is manipulated in a quantum bit (qubit).
In their paper, Grzybowski and colleagues say that research suggests that there are a large number of ways that such a spin can be manipulated (by applying successive magnetic fields, for example) such that it follows specific trajectories before returning to its original state. This could be particularly useful for creating new sequences for doing NMR or for processing quantum information. The polarization of light can also be described in terms of a point on a sphere and so the research could lead to the development of optical systems that are designed not to change the polarization of light as it is processed.
