Picture a spinning top coming to a halt. Is it possible to make it spin again and return to its original position, as if no movement had occurred? Surprisingly, mathematicians affirm that there is a universal method to revert the rotation of nearly any object.

It seems that the sole method to reverse a complicated rotation sequence is to meticulously execute the exact reverse motion, one step at a time. However, Jean Pierre Eckmann from the University of Geneva, alongside Tzvi Trusty and a research team from South Korea’s Ulsan Institute of Science and Technology (UNIST), discovered a concealed reset mechanism that modifies the initial rotation by a common scaling factor and applies this process twice.

For a spinning top, if it makes three-quarters of a turn during its first spin, you can apply an eighth scaling to retrace your steps back to the start and repeat that sequence again to achieve another quarter turn. Yet, Eckmann and Trusty have shown that this principle applies to much more intricate scenarios.

“Essentially, this property extends to nearly any rotating object, including spins, qubits, gyroscopes, and robotic arms,” Trusty explains. “You merely need to scale all rotation angles by the same factor and replicate this complex pathway twice, navigating through an intricate trajectory in space before returning to the origin.”

Their mathematical proof stems from a comprehensive catalog of all potential rotations in three-dimensional space, known as SO(3), which follows specific rules. This can be visualized as an abstract mathematical space resembling a ball. Transporting an object through various rotations in physical space translates to moving from one point to another within this ball, akin to a bug tunneling through an apple.

When a piece undergoes a complicated rotation, its corresponding trajectory in SO(3) may initiate at the center of the ball and terminate at different points within, depending on the intricacies of the rotation. The objective of reversing this rotation is akin to discovering a route back to the center, yet given that there is only one center within the ball, randomly accomplishing this is improbable.

Eckmann and Trusty realized that due to the structure of SO(3), halting a rotation midway is analogous to finding a path that ends on any point on the ball’s surface. Because the surface comprises numerous points, Trusty notes that this approach is significantly more straightforward than directly targeting the center. This insight led to a new proof.

Eckmann mentioned that they invested considerable time unraveling mathematical tensions that yielded no results. The breakthrough came from a 19th-century formula that merged the two successive rotations, known as Rodriguez’s formula, along with an 1889 theorem in number theory. Ultimately, the researchers concluded that a scaling factor is nearly always necessary for resetting.

For Eckmann, this latest research exemplifies the richness of mathematics, even in seemingly familiar domains like rotation studies. Trusty pointed out potential practical outcomes, such as in nuclear magnetic resonance (NMR), which underpins magnetic resonance imaging (MRI). Researchers assess material and tissue properties by examining the behavior of internal quantum spins under the influence of external magnetic fields. The new proof could pave the way for strategies to negate unwanted spin rotations that disrupt the imaging process.

The findings could also spur advancements in robotics, says Josie Hughes at the Federal Institute of Technology in Lausanne, Switzerland. For instance, a rolling robot may be developed to navigate a path comprising repetitive segments, featuring a reliable roll-reset-roll motion that could theoretically continue indefinitely. “Visualize a robot that could transition between any solid form and subsequently follow any desired trajectory through shape transformation,” she envisions.