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A series of rotations – a discrete walk (or continuous path) in the manifold of the rotation group SO(3) or SU(2) – can of course be inverted (starting from the end, find a walk that returns to the beginning) by performing the steps in reverse. Eckmann et alshow that, for almost all walks, there is another way: starting at the end, perform the steps in the original order (1) twice, and (2) uniformly scaled by a factor.
Apparently – I haven’t read the article – the factor depends on the walk. (One would think the abstract would say if there were.) The theorem says there exists such a factor but not how to find it. As the factor varies from 0 on up, the end point of the twice traveled path, scaled by some factor, is dense in the rotation manifold. It isn’t surprising though the fact that the end of the once traveled path (scaled) is not dense, is.
If the authors cannot give a comparatively simple way to find the factor, or at least bounds on it, the theorem isn’t of much use. It looks like there is too much hype accompanying its announcement.
The article is 5 pages and the Theorem yielding the factor is on page 4.
The article is here.
https://arxiv.org/abs/2502.14367
Sorry, but the existence of such an inversion still is interesting from a mathematical perspective. It isn't "of much use" practically without the inversion formula/calculation, but that's ok. "There exists" is still a fascinating fact.
I was immediately reminded of the anti-twist mechanism, perhaps unrelated but "reset rotation, twice/half" comes up there as well.
It's not related. The recent result states that you can pick any integer m > 1 and find a scaling factor λ for a given path such that after m repeats of that path you will return to the starting point (except for some infinitesimal number of paths that have a specific structure).
What?!
Thank you! I'm working on a robot with a very expensive slip ring, and need to send high fidelity data through it with shielding. I had no idea this was possible this will make things so much easier!
I found a related video you might find interesting.
https://www.youtube.com/watch?v=gZvimEf6DFw
I'm currently studying group theory and SO3 rotations (quaternions & matrix groups) and I'm also curious about the connection. I still have a lot to learn but I wouldn't be surprised if the reset rotation is unique, if we abstract away variation.
Damn, that's beautiful. I hope that Mr. Adams mentiond in the article got a good return from his patent.
Huh, looking just at the link at the top of the box, and forgetting the remainder of the links, this cannot work. I tried it with a flat cable. If you rotate it like that, it becomes twisted.
The final paragraph: The work could also lead to advances in robotics, says Josie Hughes at the Federal Polytechnic School of Lausanne in Switzerland. For example, a rolling robot could be made to follow a path of repeating segments, comprising a reliable roll-reset-roll motion that could, in theory, go on forever. “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.
Interestingly, that didn't come from the PR department. Hughes is a tenure-track professor whose lab builds unusual flexible robots. They're trying to use LLMs to design special-purpose grippers.[1] That's an interesting idea. Most of the cost in industrial robots is special-purpose end effector tooling. Something that could bang out a design, given "we want to put this thing in there", would be very useful.
Here are some examples of end of arm tooling.[2] Auto plants are full of this stuff, and it's all custom. An automated design system for designing all those one-off items would really speed up retooling assembly lines for a new product. Much of the research in robots involves trying to make more human-like grippers. That may be approaching the problem from the wrong end. Cheap custom tooling designed by AIs and maybe 3D printed may be the way to go.
That an LLM can do something like that is a surprise, but apparently there's been progress.
There's a YC-sized startup opportunity in this.
[1] https://www.epfl.ch/labs/create/
[2] https://eoat.net/tooling/?device=c&keyword=End Of Arm Tooling Grippers
Read the paper https://fiteoweb.unige.ch/~eckmannj/ps_files/ETPRL.pdf
Thank you! The article was "too clever" for me.
This article is written in a very annoying and misleading way. The discovery is not that rotation can be "reset". That is obvious and not surprising at all. Physical systems governed by classical mechanics are reversible just by perfectly inverting all forces, velocities, and rotations. The actual discovery is the shortcut to the original position without the need to perfectly inverse the full sequence of rotations.
Kind of like… a “hidden reset”…
> Physical systems governed by classical mechanics are reversible just by perfectly inverting all forces, velocities, and rotations
Heat is probably the best example, as even if you were able to track the movement of particles individually you'd have a very difficult time putting them back in order. The development of thermal stat-mech is one of the things that led to the quantum revolution and "new physics". But if you only have a "calculus" based understanding of physics you likely aren't going to be familiar with this. It's not much discussed (it is some) if you didn't start entering upper division physics classes or equivalent coursework. It really shows up when you get into the weeds, but understandably it isn't something stressed before then. Physics is hard enough...
Not all classical physics is time symmetric[0].
FWIW, I don't think the article is unclear. I mean they address your point in the first sentence of the second paragraph
> 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.
[note]: The real paradigm shift in quantum mechanics was that there was information that we could not access. That's what Schrodinger's Cat is about. The cat doesn't sit inside a parallel universe, a quantum superposition. It is just that there is no way to know which of the states the cat is in without opening the box. It says that we cannot have infinite precision, therefore must use statistics. So Einstein's "god doesn't play dice" comment is about that there must be some way to pull back that curtain.
For those who struggle with the pay wall: check your local library's (online) membership, it might come with the worldwide library card, which might include the New Scientist magazine.
Mine does, and therefore I can "borrow" (read for free) articles that make it to the mag.
I've been doing this for New Scientist and a few other magazines and there's always a few articles that I have found interesting that don't make it to hacker news (the whole magazine with ads comes digitally), though many of the pieces are very short half page articles that mention something new that one has to follow up on one's own for detailed information and there's regular columns like book reviews. This magazine via Libby feature is the only thing that makes me miss having an ipad or larger mobile device for reading convenience. I assume the magazine is paid for by our local library system for access so in some small way there is compensation making its way to the creators which if someone is worried about supporting them, is one way besides a subscription. (I have stopped print subscriptions because I always end up with repository of stuff I need to recycle or throw away).
In this case we can just wave bye-bye to the magazine and head to the freely available Arxiv paper they are writing about.
If you use an ad blocker, just disable inline scripts
Or just download the extension that bypasses pay walls lol
Archive of TFA:
which is reporting on the linked original publication:
https://journals.aps.org/prl/abstract/10.1103/xk8y-hycn
which has a preprint available:
https://arxiv.org/abs/2502.14367
h/t to both criddell and nicklaf who posted replies containing the above to a now [flagged][dead] comment which violates the HN guidelines, which is why I have collated this and reposted it as a top-level comment.
In future, I would advise folks who post archives and workarounds to post them as a top-level comment in addition to and/or instead doing so as replies to others, especially instead of as replies to comments that violate guidelines, as if/when those comments become [dead] for whatever (legitimate or otherwise) reason(s), their child comments also get buried except to those with showdead enabled on their profile, which requires not only an HN account and login, but also requires enabling the showdead option in one’s user profile.
From one author's home page (Jean-Pierre Eckmann - https://fiteoweb.unige.ch/~eckmannj/publications.html), the latest version of full paper:
Walks in Rotation Spaces Return Home when Doubled and Scaled (with Tsvi Tlusty) Physical Review Letters 135, 147201 (2025)
https://arxiv.org/abs/2502.14367 for the technical folks.
I don't entirely understand why they're framing rotations as so complex, outside of a play on words that I don't think they're making. Most rotations just use quaternions which are relatively simple. Their example of robotics uses quaternions and getting the inverse of any rotation is trivial - you literally just flip the signs of the 3 imaginary components of quaternions. For non-unit quaternions, you just then just renormalize the result (divide by the sum of the squares of the components).
Quaternion libraries have work to do now.
Positive potential:
Simplified “undo” mechanism: this result suggests that a given traversal (sequence of rotations) might be “reset” (i.e., returned to origin) using a simpler method than computing a full inverse sequence. That could simplify any functionality in libraries, like SpinStep[0], that deal with “returning to base orientation” or “undoing steps.”
The libraries could include a method: given a sequence of quaternion steps that moved from orientation A to orientation B, compute a scale factor λ and then apply that scaled sequence twice to go from B back to A (or A to A). This offers a deterministic “reset” style operation which may be efficient.
Orientation‐graph algorithms: in libraries used in robotics/spatial AI, the ability to reliably reset orientation (even after complex sequences) might enhance reliability of traversal or recovery in systems that might drift or go off‐course.
I must be missing something major here, but given a sequence of rotations combined into a quaternion orientation, can’t you just get the inverse rotation back to the original orientation by inverting the quaternion?
> compute a scale factor λ
The paper shows that such a factor must exist but not how to compute it. That is currently unknown and non-trivial.
I had a hard time trying to parse something understandable from the article.
This is what I got from it (I'd be happy to hear someone informed correcting me/confirming). (excerpt from a discussion yesterday I had with some friends not too math inclined)
What it seems to be the articles claim is that, you could define a scaling operation in the angles you performed, finding some constant scaling factor (say alpha) and running the operation twice to reach the identity (rotation 0 compared to baseline), e.g.:
I = R ⊕ (α.R ⊕ α.R)
In their example that would be something like (with alpha=0.3):
I = (rad(75).X ⊕ rad(20).Y ⊕ ...) ⊕ (rad(0.3x75).X ⊕ rad(0.3x20).Y ⊕ ...) ⊕ (rad(0.3x75).X ⊕ rad(0.3x20).Y ⊕ ...)
Remembering that our rotation action is non-commutative, e.g. `aX ⊕ bY != bY ⊕ aX`.
> Finding such a scaling amounts to solving a trigonometric Diophantine equation, and the solution applies to any physical system governed by SO(3) or SU(2), such as magnetic spins or qubits.
Can anyone comment on the difficulty of solving trigonometric Diophantine equations? Most of the resources I am familiar with only deal with linear or exponential versions.
I've been trying to understand as much of "maths" as I can (now enough to write that in quotes, as there isn't a "single" maths) and still a layman, I love reading about discoveries like these, and the fact that you still can have discoveries in things thought to be so fundamental..
Wish they showed a picture of both. A path over time that changes color and two paths combined to recreate it.
Any implications for MRI/ NMR here? The basis of arguably most pulse sequences is undoing rotation in some way, it’s not immediately obvious if this finding could provide any new refocusing sequences.
This made me wonder if there are knots you can't untangle.
this doesn't seem very difficult of a result to me; an arbitrary rotation is a move from one endpoint to each other on SO(3) wnich is double-covered by SU(2) ≅ ³; wiog consider the path between endpoints a geodesic then o course two (or even one) appropriately-scaied copies of the originaL rotation will suffice to revert it
> Mathematicians thought that they understood how rotation works, but now a new proof has revealed a surprising twist
Clever intro.
How does this help solve a rubix cube?
Does anyone have a link to research itself? I don’t want to sign up to “new scientist” to see behind the sign up screen to see if they included a link or not
This article leads to a paywall where I am, making it of no use. Perhaps someone else has done a better job of summarizing the paper elsewhere, and that should be posted instead?
Reminds me of belt/plate trick and anti-twister mechanism.
The belt trick / plate trick / Dirac's string trick is nicely demonstrated in below video: https://m.youtube.com/watch?v=EgsUDby0X1M
https://en.wikipedia.org/wiki/Plate_trick
In mathematics and physics, the plate trick, also known as Dirac's string trick (after Paul Dirac, who introduced and popularized it), the belt trick, or the Balinese cup trick (it appears in the Balinese candle dance), is any of several demonstrations of the idea that rotating an object with strings attached to it by 360 degrees does not return the system to its original state, while a second rotation of 360 degrees, a total rotation of 720 degrees, does.
https://en.wikipedia.org/wiki/Anti-twister_mechanism
The anti-twister or antitwister mechanism is a method of connecting a flexible link between two objects, one of which is rotating with respect to the other, in a way that prevents the link from becoming twisted. The link could be an electrical cable or a flexible conduit.
This mechanism is intended as an alternative to the usual method of supplying electric power to a rotating device, the use of slip rings. The slip rings are attached to one part of the machine, and a set of fine metal brushes are attached to the other part. The brushes are kept in sliding contact with the slip rings, providing an electrical path between the two parts while allowing the parts to rotate about each other.
However, this presents problems with smaller devices. Whereas with large devices minor fluctuations in the power provided through the brush mechanism are inconsequential, in the case of tiny electronic components, the brushing introduces unacceptable levels of noise in the stream of power supplied. Therefore, a smoother means of power delivery is needed.
A device designed and patented in 1971 by Dale A. Adams and reported in The Amateur Scientist in December 1975, solves this problem with a rotating disk above a base from which a cable extends up, over, and onto the top of the disk. As the disk rotates the plane of this cable is rotated at exactly half the rate of the disk so the cable experiences no net twisting.
What makes the device possible is the peculiar connectivity of the space of 3D rotations, as discovered by P. A. M. Dirac and illustrated in his Plate trick (also known as the string trick or belt trick). Its covering Spin(3) group can be represented by unit quaternions, also known as versors.
https://en.wikipedia.org/wiki/3D_rotation_group
In mechanics and geometry, the 3D rotation group, often denoted SO(3), is the group of all rotations about the origin of three-dimensional Euclidean space R³ under the operation of composition.
By definition, a rotation about the origin is a transformation that preserves the origin, Euclidean distance (so it is an isometry), and orientation (i.e., handedness of space). Composing two rotations results in another rotation, every rotation has a unique inverse rotation, and the identity map satisfies the definition of a rotation. Owing to the above properties (along composite rotations' associative property), the set of all rotations is a group under composition.
Does it work for brakes?
up next, unscrambling an egg!
Doesn’t this sound like a sneaky way for a mathematician to work on time travel?