Mathematicians have found a hidden ‘reset button’ for undoing rotation

Picture a spinning top gradually coming to a stop. Is there a technique to rotate it again and return it to its original position, as if it had never been spun? To many’s surprise, mathematicians claim there exists a universal method to reverse the rotation of almost any object.

Commonly, one might think that to counteract a complex series of rotations, you’d have to meticulously execute the inverse movements in sequence. However, Jean-Pierre Eckmann from the University of Geneva and Tsvi Tlusty from the Ulsan National Institute of Science and Technology (UNIST), South Korea, have uncovered a concealed reset mechanism involving adjusting the magnitude of the initial rotation by a common scale factor, a technique known as scaling, and applying it twice.

For the spinning top specifically, if your initial rotation twisted it three-quarters, you can revert to the original position by scaling the rotation to one-eighth and repeating it to achieve an additional quarter rotation. Eckmann and Tlusty have demonstrated that this method extends to far more intricate scenarios.

“This principle is essentially a characteristic of nearly any rotating object, such as a qubit, spin, gyroscope, or even a robotic limb,” Tlusty explains. “When these objects trace a highly intricate path through space, simply scaling all rotation angles uniformly and retracing that complex route twice enables a return to the starting point.”

Their mathematical foundation begins with a comprehensive list of all feasible rotations in three-dimensional space. This collection, termed SO(3), can be represented through an abstract mathematical structure akin to a ball, where maneuvering an object through a series of rotations in the physical world equates to navigating from one point within this ball to another, similar to a worm tunneling through an apple.

When a top is spun in a complex manner, the corresponding trajectory within the SO(3) space begins at the ball’s center and may end at any point on its surface, contingent on the specifics of the rotation. The challenge of reversing the rotation parallels the search for a path leading back to the center, and given only one center, the chances of randomly achieving this are quite minimal.

What Eckmann and Tlusty discovered is that, due to the structure of SO(3), reversing a rotation halfway is similar to identifying a pathway that takes you to any point on the surface of the ball. This is a far more attainable task than aiming to reach the center, given that the surface is composed of numerous points, according to Tlusty. This insight was pivotal in their proof.

The duo encountered numerous dead ends while exploring various mathematical theories, explains Eckmann. Ultimately, they relied on a 19th-century formula for combining two rotations known as the Rodrigues formula, and a theorem from 1889 associated with number theory. They concluded that the scaling factor required for their reset mechanism is almost always attainable.

Eckmann views their findings as an illustration of the rich landscape of mathematics even in the well-researched domain of rotation studies. Tlusty adds that their work could have practical implications, for instance, in nuclear magnetic resonance (NMR), which underpins magnetic resonance imaging (MRI). Here, scientists assess the characteristics of materials and biological tissues by examining how quantum spins respond to externally applied rotations from magnetic fields. Their newfound proof may aid in formulating strategies to reverse unwanted spin rotations that could disrupt imaging procedures.

The implications could extend into robotics, according to Josie Hughes from the Federal Polytechnic School of Lausanne. She speculates that a rolling robot could potentially trace a path of repetitive segments establishing a dependable roll-reset-roll cycle with the potential to continue indefinitely. “Imagine a robot capable of transforming between any solid shape; it could effortlessly navigate any desired trajectory through its shape-shifting abilities,” she envisions.