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

114 pointsposted 6 days ago
by mikhael
(newscientist.com)

70 Comments

AugusteDupin

2 hours ago

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.

v7n

10 hours ago

I was immediately reminded of the anti-twist mechanism, perhaps unrelated but "reset rotation, twice/half" comes up there as well.

https://en.wikipedia.org/wiki/Anti-twister_mechanism

dandanua

9 hours ago

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).

Syntonicles

8 hours ago

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.

AugusteDupin

2 hours ago

As meindnoch points out, the connection needs to loop over the rotating object. That is no problem if the only affect of the rotation that interests you is the centrifugal force.

When you give plasma (not whole blood) the nurses use a centrifuge machine that seems impossible: one tube goes from you to it (carrying whole blood), another tube goes from it back to you (carrying plasma depleted blood). The mechanism of Dale. A. Adams keeps the tubes from twisting. Search “antitwister mechanism patent” for a drawing of the mechanism. As for the principle behind the mechanism, see http://Antitwister.ariwatch.com for a PC program where you can adjust every variable imaginable.

meindnoch

7 hours ago

There's a bit of a caveat with the anti-twister mechanism, namely, that the wiring must be loose enough to pass around the supplied rotating part.

SyzygyRhythm

6 hours ago

This is important. The mechanism doesn't really work the way you want most of the time. I occasionally see a claim that you can power a carousel with this method, but it doesn't work. You would have to have the cable go out and around the carousel structure, and then into the top. And the cable would still move relative to the ground and the carousel.

You could, in principle, have a totally internal system, but with arms that grab and release the cable at intervals so that the looped portion can pass by them. You could arrange the timing so that electrical contact is never lost. But you are still making/breaking contact and it starts to lose some apparent advantages compared to a slip ring.

That's not to say it isn't still useful for some purposes, like maybe a radio antenna that isn't too impacted by a cable moving in front on occasion. But it doesn't eliminate all uses for a slip ring.

jojobas

6 hours ago

And no axle to rotate on.

v7n

7 hours ago

Always happy to share! I came across this while planning a 3D scanning (photogrammetry) rig. Perhaps you'll be the one to figure out gravity can be modelled as a rotation around an axis in a fourth dimension, wrapping clingy spacetime around itself? ;) I'm not clever enough for that.

crooked-v

8 hours ago

Damn, that's beautiful. I hope that Mr. Adams mentiond in the article got a good return from his patent.

kronodeus

7 hours ago

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.

clysm

6 hours ago

Kind of like… a “hidden reset”…

kronodeus

5 hours ago

The title itself is not the problem, although even that is sensationalized. I was referring to the contents of the article, which have statements like this:

"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..."

Which, as an introduction, just misses the mark completely by highlighting the least surprising possible interpretation of the research.

godelski

6 hours ago

  > Physical systems governed by classical mechanics are reversible just by perfectly inverting all forces, velocities, and rotations
This doesn't really make sense. To do that you'd have to end up bringing in quantum dynamics, and well... we know how that goes.

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.
[0] There are examples on this page that do not require relativity or quantum mechanics, even though some do. https://en.wikipedia.org/wiki/T-symmetry

[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.

oh_my_goodness

5 hours ago

Comment refers to classical mechanics, not all of classical physics and explicitly not quantum mechanics.

godelski

4 hours ago

  > refers to classical mechanics
Thermodynamics is, in fact, part of classical mechanics

oh_my_goodness

4 hours ago

It is not. That’s exactly what it isn’t.

godelski

3 hours ago

I think you're confusing thermal dynamics with quantum mechanics.

kronodeus

5 hours ago

Not talking about thermodynamics here. The discovery referenced in this article also does not solve for thermodynamics or entropy.

And yes, you're right, the article does mention this later. I'm still bothered by the sensationalized introduction and title.

godelski

4 hours ago

  > Not talking about thermodynamics here.
My mistake, you said "Classical Mechanics", so I took it as such.

But thermodynamics is not required either. Chaos theory would be of important note here. Take the double pendulum for example. It is a chaotic function because unless you have the initial state you cannot make accurate predictions as to its forward time evolution. This is a deterministic system because there is no randomness in the forward time evolution. But it is chaotic because it is sensitive to initial conditions. I think you can see that there's a careful choice of words here and that once we start trying to reverse the evolution we will not be able to do so. We have to deal with injective functions and I'm not sure many people really think P=NP. Just because f(t) has a unique map doesn't mean f^-1(t) does. Do not confuse "deterministic" with "predictable" nor "invertible" (nor "reversible" and "invertible"). Nor should you confuse "Newtonian Mechanics" with "Classical Mechanics".

Besides, I don't think you can throw out thermodynamics just so easily. With it you throw out many things like friction too. Not to mention that you're suggesting you're also throwing out fluid mechanics. For the fun of it, let me introduce you to Norton's dome since we might want to look at determinism in Newtonian Mechanics and a frictionless system ;)

kmarc

9 hours ago

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.

stevenwoo

9 hours ago

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).

kazinator

8 hours ago

In this case we can just wave bye-bye to the magazine and head to the freely available Arxiv paper they are writing about.

viciousvoxel

7 hours ago

If you use an ad blocker, just disable inline scripts

groby_b

7 hours ago

Or, you know, you could use a mechanism that actually guarantees them some revenue and doesn't just burn the publication to the ground because you feel entitled to free access.

typpilol

9 hours ago

Or just download the extension that bypasses pay walls lol

voxleone

8 hours ago

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.

[0] https://github.com/VoxleOne/SpinStep

ginko

7 hours ago

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?

cormacrelf

6 hours ago

You can absolutely do that and there is nothing for general linear algebra libraries to do.

The actual paper is very clear about what it's for: https://fiteoweb.unige.ch/~eckmannj/ps_files/ETPRL.pdf

It says:

    Consider now a general time-dependent field B(t) of duration T. The pulse B(t) may be extremely convoluted ... Can one make the field B(t) return the system to its original state at the end of the pulse...?
This pulse is modelled as a long sequence of rotations. For maths purposes if you had such a sequence, you can obviously just multiply all the rotations together and find the inverse very easily. For physics purposes, you don't really have access to each individual rotation, all you can do is tune the pulse. Creating an "inverse pulse" is quite unwieldy, you might literally need to create new hardware. The paper asks "what if we just amplified the pulse? Can we change this alone and make it not impart any rotation?"

They are trying to take any pulse B(t) and zero out any rotation it imparts on some particle or whatever by

    uniformly tuning the field’s magnitude, B(t) → λB(t) or by uniformly stretching or compressing time, B(t) → B(λt)
And the answer is that you can do that, but you might have to perform the pulse twice.

ogogmad

6 hours ago

I think even conjugating it. The formula for rotation via quats is v->qvq^{-1} = qvq^*/|q|^2.

meindnoch

7 hours ago

>using a simpler method than computing a full inverse sequence

What are you even talking about? Rotations form a group. Any orientation "A" can be reached from any other orientation "B" with a single rotation. It's an O(1) operation. Always has been. What you wrote makes no sense whatsoever.

the__alchemist

6 hours ago

#1: BM.

#2: His point is that this could be applied compute that single rotation.

meindnoch

6 hours ago

Makes no sense. Computing the rotation between any two orientations (represented as quaternions) is simply a matter of dividing one quaternion by the other. It's an O(1) operation. It's a non-problem.

badosu

2 hours ago

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`.

SoftTalker

an hour ago

> Mathematicians thought that they understood how rotation works, but now a new proof has revealed a surprising twist

Clever intro.

nyrikki

6 hours ago

> 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.

andy99

7 hours ago

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.

vntok

6 hours ago

> For Eckmann, the new work is a showcase of how rich mathematics can be even in a field as well-trod as the study of rotations. Tlusty says that it could also have practical consequences, for instance, in nuclear magnetic resonance (NMR), which is the basis of magnetic resonance imaging (MRI). Here, researchers learn 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 proof could help develop procedures for undoing unwanted spin rotations that would interfere with the imaging process.

koolala

10 hours ago

Wish they showed a picture of both. A path over time that changes color and two paths combined to recreate it.

dukeofdoom

8 hours ago

This made me wonder if there are knots you can't untangle.

jonathrg

5 hours ago

Every (mathematical) knot is one that can't be untangled, by definition.

Every knot with a cut can be trivially collapsed go a point by moving one of the endpoints to the other one through the path of the knot

ekunazanu

7 hours ago

Yup, the trefoil knot is one

eightys3v3n

7 hours ago

I'm not sure I follow. Every knot is defined as if you close the ends it cannot be unravelled without cutting the ends again. So the trifoil knot is included in this... but so is almost if not every other knot aren't they? Do we have "knots" that aren't mathematical? I feel like if you tie any "knot" then fix the ends together most or all of them would not be possible to untangle.

ineedasername

9 hours ago

Doesn’t this sound like a sneaky way for a mathematician to work on time travel?

swader999

9 hours ago

Baby steps, first is the roulette table.

alphan0n

8 hours ago

> Often dabo girls were specifically instructed by their employers to distract players into losing. A common saying in dabo was "Watch the wheel, not the girl."

echelon

9 hours ago

Kardashev Type III civilization:

Reverse the light cone, resimulate all moments of the past down to the neurotransmitter level. The thoughts, feelings, and memories locked inside your head.

From Neanderthal to Shakespeare to you, we could bring back everyone who has ever lived and put them in a theme park without any of them ever even knowing.

Some simulation instances might be completely accurate. For historians or as a kind of theme park or zoo.

Maybe that's us right now.

Some simulation instances might be for entertainment. They might resemble plain and ordinary, mundane day to day life (like this very moment), and then all of a sudden dramatically morph into a zombie monster outbreak tornado asteroid alien invasion simulator.

Or maybe it's obvious when a group of future gamer nerds log into an instance to role play Musk and Zuckerberg and Altman and speed run "winning". Or try to get a "high score".

Maybe it'll be eternal heaven - just gifted to us without reason or cause. That'd be nice.

Or perhaps and seemingly more likely, a bunch of sadomasochistic hell sims for psychopaths. Where some future quadrillionaire beams up into the matrix to torture poor people that used to live just for fun. It's not like we would have any rights or protections or defense against it.

Who knows.

sebastiennight

7 hours ago

1 - A copy of me is not me.

2 - There might be a form of hubris in thinking that replicating a conscious person by copying all their neurotransmitters is enough to have a continuity between the original and the copy.

It can be easily evidenced if you consider that the people who tend to believe this, will have a level of granularity in their beliefs that depend on their era and their own knowledge, so maybe a century ago you'd think copying the nerve/neuron arrangement would be enough, and a few decades later someone would've said that you need the exact arrangement of molecules or atoms, while maybe in 2025 we'd be thinking in terms of electron clouds or quarks.

But to think that today we have finally arrived at a complete and final understanding of the basic blocks and surely, there is no possible finer understanding that would make our current view quaint in the eyes of a person from 2085 is the hubris I'm talking about.

echelon

7 hours ago

> enough to have a continuity

Who said continuity mattered? How would a copy or original know which they were? Does it even matter?

How would you even know you were in a simulation? We seemingly don't have the tools to know.

Whatever the case, if you're the copy in the hell simulator getting thrown into the meat grinder, I don't really think the distinction of "original vs copy" is the most pressing issue.

> while maybe in 2025 we'd be thinking in terms of electron clouds or quarks.

We can't fathom what level of control over the physical world an advanced intelligence might have. Maybe they can create entire universes. Maybe there are structures and dimensions beyond our understanding. I don't know and can't reason about them, but I'm willing to prescribe them god powers on account of the fact I have no idea.

Maybe our logic and intuition, tools like Occam's Razor, are fixed to an artificial distribution of event occurrences that is entirely constructed. Perhaps not unlike the fundamental constants of the universe. We wouldn't know any differently.

None of this is not measurable. Indistinguishable from fantasy.

DonHopkins

6 hours ago

Check out Surface Detail by Iain M. Banks:

https://en.wikipedia.org/wiki/Surface_Detail

>Each chapter of the book covers one or more of the six main protagonists—Lededje Y'breq, a chattel slave; Joiler Veppers, an industrialist and playboy; Gyorni Vatueil, a soldier; Prin and Chay, Pavulean academics; and Yime Nsokyi, a Quietus agent. Some of the plot occurs in simulated environments. As the book begins, a war game—the "War in Heaven"—has been running for several decades. The outcome of the simulated war will determine whether societies are allowed to run artificial Hells, virtual afterlives in which the mind-states of the dead are tortured. The Culture, fiercely anti-Hell, has opted to stay out of the war while accepting the outcome as binding.

TomasNieteriter

7 hours ago

In Ernst Mach's Opera Omnia, his Principia had a `gedenken experiment' visiting a related question about angular inertia, as an affection of all the matter in the universe and its simultaneity with local causation. He inferred by simile of unwinding the trajectory of a toy spinning top on the possibility of reversing the arrow of time.

aspenmayer

8 hours ago

Archive of TFA:

https://archive.is/08ig5

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.

Razengan

9 hours ago

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..

jkrshnmenon

6 hours ago

I'm also trying to understand the implications of this work.

Does it imply that some for some functions F(x) = y, you can compute x given the value of y without computing the inverse of F ?

If so, what constraints does F need to meet for this ?

dist-epoch

8 hours ago

Neat factoid: there is something special about rotations in 3D. They are not "simply-connected", which means that there are 2 distinct classes of rotations. And this property is deeply important in quantum physics.

dandanua

8 hours ago

It's a bit more complicated than "2 classes of rotations", though there is magic indeed. I've tried to explain it in this post https://dandanua.github.io/2021/08/23/the-spin-of-a-human-bo...

alanbernstein

an hour ago

Thanks for sharing. I'm very familiar with the basic mechanics of quaternion rotation, and I've been interested in a deeper understanding of this double-cover concept, but I just don't get it. I've seen the belt trick and it feels more like an illusion than an illustration of some deep truth.

I like how you've connected it to spin, but I still don't understand how that is a real physical property rather than a mathematical artifact.

I don't quite grasp the significance of your "different look". Can you suggest any other reading?

tonijn

5 days ago

Does it work for brakes?