Researchers have discovered a universal way to reverse complex rotations of objects by scaling the motion and repeating it twice. This finding, applicable to spins, qubits, and robotic arms, relies on properties of three-dimensional rotation space. The proof could aid fields like medical imaging and robotics.
Imagine spinning a top through a series of twists and wanting to return it precisely to its starting position without retracing each step exactly. Mathematicians Jean-Pierre Eckmann at the University of Geneva in Switzerland and Tsvi Tlusty at the Ulsan National Institute of Science and Technology in South Korea have proven that this is possible for nearly any rotating object.
Their method involves scaling all rotation angles by a common factor and repeating the scaled sequence twice, effectively resetting the object to its origin. For instance, if a top is rotated by three-quarters of a turn, scaling it to one-eighth and repeating twice adds the necessary quarter turn to complete the reversal.
The proof draws on the structure of SO(3), the mathematical space of all possible rotations in three dimensions, which resembles a ball. A complex rotation path starts at the ball's center and ends elsewhere; undoing it means returning to the center. The researchers found that scaling and repeating twice exploits the ball's geometry, where halfway reversal lands on the surface—a vast set of points easier to target than the single center.
Eckmann and Tlusty combined the 19th-century Rodrigues formula for successive rotations with an 1889 number theory theorem to show the scaling factor almost always exists, after exploring many dead-end paths.
"It is actually a property of almost any object that rotates, like a spin or a qubit or a gyroscope or a robotic arm," says Tlusty. "If [objects] go through a highly convoluted path in space, just by scaling all the rotation angles by the same factor and repeating this complicated trajectory twice, they just return to the origin."
Practical implications include nuclear magnetic resonance (NMR), foundational to MRI, where the technique could correct unwanted spin rotations during imaging. In robotics, Josie Hughes at the Federal Polytechnic School of Lausanne suggests it enables endless roll-reset sequences for rolling or morphing robots. "Imagine if we had a robot that could morph between any solid body shape, it could then follow any desired path simply through morphing of shape," she says.
The work appears in Physical Review Letters (DOI: 10.1103/xk8y-hycn), highlighting mathematics' depth in familiar areas like rotations.