Mathematicians have found a hidden 'reset button' for undoing rotation

Summary

Can you undo a spinning top?Shutterstock Imagine spinning a top and then letting it come to rest. Is there a way for you to spin the top again so it ends up in the exact position it started, as if you had never spun it at all? Surprisingly, yes, say mathematicians who have discovered a universal recipe for undoing the rotation of nearly any object. Intuitively, it feels like the only way to undo a complicated sequence of rotations is by painstakingly doing the exact opposite motions one by one. But 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. In the case of the spinning top, if your initial rotation had turned the top by three-quarters, you can return to the start by scaling your rotation to one-eighth, then repeating it twice to give you an extra quarter rotation. But Eckmann and Tlusty have shown it is also possible to do this for far more complicated situations. “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.” Their mathematical proof starts with a catalogue of all rotations that are possible in three spatial dimensions. This catalogue, known as SO(3), can be described using an abstract mathematical space that has special rules and is structured like a ball, with the act of pushing an object through a sequence of rotations in real space corresponding to moving from one point within the ball to another, like a worm tunnelling through an apple. When you spin a top in some complicated way, the equivalent path within ...