by … Jean-Pierre Eckmann at the University of Geneva in Switzerland and Tsvi Tlusty at the Ulsan National Institute of Science and Technology (UNIST) in South Korea have found a hidden reset button that involves changing the size of the initial rotation by a common factor, a process known as scaling, and repeating it twice.
Mathematicians represent rotations using a space called SO(3) — a three-dimensional map where every point corresponds to a unique orientation. At the very center lies the identity rotation: the object’s original state. Normally, retracing a complex path through this space wouldn’t bring you back to that center. But Eckmann and Tlusty found that scaling all rotation angles by a single factor before repeating the motion twice acts like a geometric reset.
So for example:
- If your first rotation sequence tilted the object 75 degrees this way, 20 degrees that way, and so on, you could shrink all those angles by, say, a factor of 0.3, and then run that shortened version two times in a row.
- After those two runs, the object returns perfectly to its starting position — as if nothing had ever happened.
In their proof, the researchers blended a 19th-century tool for combining rotations (Rodrigues’ rotation formula) with Hermann Minkowski’s theorem from number theory. Together, these revealed that “almost every walk in SO(3) or SU(2), even a very complicated one, will preferentially return to the origin simply by traversing the walk twice in a row and uniformly scaling all rotation angles.”
