> Is it possible to cancel out a complicated spin without painstakingly reversing every single move? Surprisingly, the answer is yes.
> Mathematicians Jean-Pierre Eckmann (University of Geneva) and Tsvi Tlusty (UNIST, South Korea) recently proved that almost any object—whether it’s a spinning top, a tumbling satellite, a twisted protein, or even a scrambled Rubik’s Cube—has a hidden “reset button” for its orientation.
> Instead of undoing the motion step by step in reverse order, you can take the entire original sequence of rotations, scale it by a certain constant factor (make every turn bigger or smaller by the same proportion), perform that scaled version once, then do it again—and the object snaps perfectly back to its starting orientation. Two scaled copies of the same motion are enough to erase it completely.
> It feels deeply counterintuitive. We’re used to thinking that rotations in 3D space don’t commute and that the only safe way to return home is to retrace your path exactly backward. Yet this new result reveals a previously unknown geometric symmetry: certain scaling factors turn the rotation group into something that has a kind of built-in “double and cancel” feature.
> The discovery applies to any rigid body moving in three dimensions and may simplify algorithms in robotics (for reorienting a robot arm without tracking every prior move), computer graphics, molecular dynamics simulations, spacecraft attitude control, and even some problems in quantum mechanics.
> In short, nature has been hiding a remarkably simple trick: sometimes the fastest way to undo a complex dance of spins isn’t to moonwalk backward through every step; it’s to perform an enlarged (or shrunken) version of the same dance twice.
nomilk•2mo ago
> Mathematicians Jean-Pierre Eckmann (University of Geneva) and Tsvi Tlusty (UNIST, South Korea) recently proved that almost any object—whether it’s a spinning top, a tumbling satellite, a twisted protein, or even a scrambled Rubik’s Cube—has a hidden “reset button” for its orientation.
> Instead of undoing the motion step by step in reverse order, you can take the entire original sequence of rotations, scale it by a certain constant factor (make every turn bigger or smaller by the same proportion), perform that scaled version once, then do it again—and the object snaps perfectly back to its starting orientation. Two scaled copies of the same motion are enough to erase it completely.
> It feels deeply counterintuitive. We’re used to thinking that rotations in 3D space don’t commute and that the only safe way to return home is to retrace your path exactly backward. Yet this new result reveals a previously unknown geometric symmetry: certain scaling factors turn the rotation group into something that has a kind of built-in “double and cancel” feature.
> The discovery applies to any rigid body moving in three dimensions and may simplify algorithms in robotics (for reorienting a robot arm without tracking every prior move), computer graphics, molecular dynamics simulations, spacecraft attitude control, and even some problems in quantum mechanics.
> In short, nature has been hiding a remarkably simple trick: sometimes the fastest way to undo a complex dance of spins isn’t to moonwalk backward through every step; it’s to perform an enlarged (or shrunken) version of the same dance twice.
Source: https://x.com/Rainmaker1973/status/1996635202605756658