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

Imagine spinning a top and then letting it stop. Is there a way for you to spin the top again so that it ends up in the exact position it started in, as if you never spun it at all? Surprisingly, yes, say mathematicians who have discovered a universal recipe for canceling the rotation of almost all objects.

Intuitively, it seems that the only way to undo a complicated sequence of rotations is to painstakingly perform the exact opposite movements, one at a time. But Jean-Pierre Eckmann of the University of Geneva in Switzerland and Tsvi Tlusty of the Ulsan National Institute of Science and Technology (UNIST) in South Korea discovered 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 top, if your initial spin had the top spinning three-quarters, you can go back to the start by increasing your spin to one-eighth, then repeating it twice to give you an extra quarter-spin. But Eckmann and Tlusty showed that it was also possible to do this in much more complicated situations.

“It’s actually a property of almost every object that rotates, like a spin, a qubit, a gyroscope or a robotic arm,” says Tlusty. “If [objects] travel a very convoluted path in space, simply by scaling all rotation angles by the same factor and repeating this complicated trajectory twice, they simply return to the origin.

Their mathematical proof begins with a catalog of all possible rotations in three spatial dimensions. This catalog, known as SO(3), can be described using an abstract mathematical space with special rules and structured like a ball, with the action of pushing an object through a sequence of rotations in real space corresponding to moving from one point on the ball to another, like a worm tunneling through an apple.

When you spin a top in a complicated way, the equivalent path in SO(3) space starts at the very center of the ball and can end at any other point on the ball, depending on the details of the spin. The goal of unspinning is equivalent to finding a path back to the center of the ball, but since there is only one center, your chances of doing this randomly are slim.

What Eckmann and Tlusty realized is that, because of the way SO(3) is structured, canceling a rotation halfway is equivalent to finding a path that will take you anywhere on the surface of the ball. This is much easier than trying to reach the center, because the surface is made up of many points, Tlusty explains. This was the key to the new evidence.

The two men spent a lot of time researching mathematical reasoning that led nowhere, Eckmann says. What ultimately worked was a 19th-century formula for combining two subsequent rotations, called the Rodrigues formula, and an 1889 theorem from a branch of mathematics known as number theory. Ultimately, the researchers concluded that the scale factor needed to reset them almost always exists.

For Eckmann, this new work illustrates how rich mathematics can be, even in a field as widespread as the study of rotations. Tlusty says this could also have practical consequences, for example in the field of nuclear magnetic resonance (NMR), which is the basis of magnetic resonance imaging (MRI). Here, researchers learn the properties of materials and tissues by studying the response of quantum spins inside them to rotations imposed on them by external magnetic fields. The new evidence could help develop procedures to cancel out unwanted spin rotations that could interfere with the imaging process.

This work could also lead to advances in robotics, says Josie Hughes of the École Polytechnique Fédérale in Lausanne, Switzerland. For example, a rolling robot might be required to follow a trajectory of repetitive segments, including reliable roll, reset, and roll motion that could, in theory, last indefinitely. “Imagine if we had a robot that could transform into any solid body shape, then it could follow any desired path just by changing shape,” she says.