6 ms·

'Islands' of regularity discovered in the famously chaotic three-body problem

119 points by wglb 2y ago

big-green-man 2y ago
I'd bet a thousand dollars that when graphed, these islands of regularity will correspond to the graph of the Mandelbrot set.
- eszed 2y ago
  Why Mandelbrot, specifically?
  - big-green-man 2y ago
    I am going out on a limb and speculating that these stable sets of conditions correspond to Julia sets by some mathematical relationship and that the underlying fundamental principle is the same. https://en.m.wikipedia.org/wiki/Julia_set https://en.m.wikipedia.org/wiki/Julia_set
    - m3047 2y ago
      The Mandelbrot and Julia sets are the outcomes of specified functions and have a relationship with each other. But yeah I would speculate that something similar, in the way that a Mandelbrot set is self-similar, would be observed in at least some circumstances. Gave you a vote for that. Edit: reading the article I can see that Julia sets are defined separately from the Mandelbrot set and the corresponding Julia set(s) are an example. So maybe you're right. I'm not really a mathematician, I just use math to get stuff done.
- moi2388 2y ago
  I’m willing to take that bet.
  - big-green-man 2y ago
    So what do you think the math is going to look like about the starting values of these stable states? I'm speculating that they correspond to Julia sets by some fundamental relationship.
    - rcxdude 2y ago
      1) They're not stable, they're regular. The point is these areas of the state space have similar outcomes if they are perturbed slightly, as opposed to the result being random 2) there's literally a plot of it in the article, and while it shares some properties with the mandelbrot set (symmetry about the X axis, is a fractal), it doesn't resemble it at all.
- SideburnsOfDoom 2y ago
  Will it be that fractal? it seems tenuous. Will it be a fractal? That would make sense.
- pixelpoet 2y ago
  I've spent a good portion of my life working on fractal software, trying basically every fractal rendering algorithm there is, reading everything I can about it, managing fractal art communities and organising meetups... so I'm obviously very pro-fractals biased, and even so, I'm curious what makes you conjecture so strongly that 3-body stability islands would have anything to do with the Mandelbrot set. Why that one in particular, over the vast infinitude of possible fractal shapes?
didgeoridoo 2y ago
Does anyone know if that visualization is representative or just an illustration? I’m struggling with how all these parameters (mass, speed, and relative direction for all three objects) could map to a fractal on a 2 dimensional plane.
- rcxdude 2y ago
  The study didn't plot the entire state space. It's simply a map of the end state of a limited space of parameters (specifically, the mass of the objects is fixed, two of them are in a particular orbit, and basically only the angle and timing of when the third object approaches is varied). The plot just shows which object gets thrown out of the system for each initial configuration.
  - didgeoridoo 2y ago
    That makes a huge amount more sense. Thanks!
oolonthegreat 2y ago
AFAIK, these islands of regularity/stability are a common aspect of chaotic systems, say the logistic map[1]. [1]: https://en.m.wikipedia.org/wiki/Logistic_map https://en.m.wikipedia.org/wiki/Logistic_map
- Rallen89 2y ago
  Off topic buying haven't seen a bit of code in Wikipedia to run to see a visualisation before
- ddejohn 2y ago
  I may be mistaken but I seem to recall having also read on numerous occasions that ejection of one of the bodies is an inevitability in these types of systems. Given the limited state space of the simulation, I'm not sure I see what the big discovery is here. It's certainly a neat result to see it visualized though.
pkoird 2y ago
I never really understood the 3 body problem. Am I correct in saying that given three bodies that attract one another, there is currently no known way of predicting, at arbitrary time, where they'll end up at, even though it is possible to simulate to reasonable degree of accuracy for a limited time horizon? If that is the case, is this an example of what Stephen Wolfram regularly calls as computationally irreducible? Also it is so interesting to me that the jump from 2 to 3 in different domains herald such unfathomable complexity. 2SAT for instance is polynomially solved whereas 3SAT is NP Hard. If I were a competent mathematician, I'd definitely investigate the link between NP Hardness and Computationally irreducible chaotic process like 3Body problem.
- nightowl_games 2y ago
  Yes, I believe it is the same as Wolfram's "Computational Irreducability". I'll copy paste the question I emailed him about it: > In computers, floating point numbers have a limited precision, based on how many bits are used. Does the innate 'error' in floating point numbers have implications on your theory of 'Computational Irreducability'? Are the 'errors' introduced with each floating point operation is fundamentally computationally irreducable? Does this have a domino effect that causes basic classical mechanics to become computationally irreducible? Like the integration of velocity into position, it is a O(1) operation to compute the future position given a velocity and a starting point, but if a computer is rendering each individual frame, it will accumulate error in each frame, and the final frame may end up in a different position. Does this concept relate to the real world through things like the Planck length? Is space perhaps not purely continuous?
  - ramesh31 2y ago
    >Is space perhaps not purely continuous? Pretty sure that's the entire basis of quantum mechanics
    - metalliqaz 2y ago
      space? or energy?
pradn 2y ago
The paper is at [1], and the summary chart is at [2]. The top chart shows which of the three bodies escapes the system given initial starting conditions. The bottom one makes it easier to see patterns by reducing noise. It uses the k-nearest-neighbors algorithm to find the dominant color near each pixel. More than the large pools of stability, what's interesting are the bands of stability - like electromagnetic fields. I have no idea how one codes this up in a computer, with fixed precision math and all. Numerical methods is a black box for me. [1]: https://doi.org/10.1051/0004-6361/202449862 https://doi.org/10.1051/0004-6361/202449862 [2]: https://www.aanda.org/articles/aa/full_html/2024/09/aa49862-24/F3.html https://www.aanda.org/articles/aa/full_html/2024/09/aa49862-...
jmpwat 2y ago
Rehydrate! Rehydrate!

119 points by wglb 2y ago

I'd bet a thousand dollars that when graphed, these islands of regularity will correspond to the graph of the Mandelbrot set.

eszed 2y ago
Why Mandelbrot, specifically?
- big-green-man 2y ago
  I am going out on a limb and speculating that these stable sets of conditions correspond to Julia sets by some mathematical relationship and that the underlying fundamental principle is the same. https://en.m.wikipedia.org/wiki/Julia_set https://en.m.wikipedia.org/wiki/Julia_set
  - m3047 2y ago
    The Mandelbrot and Julia sets are the outcomes of specified functions and have a relationship with each other. But yeah I would speculate that something similar, in the way that a Mandelbrot set is self-similar, would be observed in at least some circumstances. Gave you a vote for that. Edit: reading the article I can see that Julia sets are defined separately from the Mandelbrot set and the corresponding Julia set(s) are an example. So maybe you're right. I'm not really a mathematician, I just use math to get stuff done.
moi2388 2y ago
I’m willing to take that bet.
- big-green-man 2y ago
  So what do you think the math is going to look like about the starting values of these stable states? I'm speculating that they correspond to Julia sets by some fundamental relationship.
  - rcxdude 2y ago
    1) They're not stable, they're regular. The point is these areas of the state space have similar outcomes if they are perturbed slightly, as opposed to the result being random 2) there's literally a plot of it in the article, and while it shares some properties with the mandelbrot set (symmetry about the X axis, is a fractal), it doesn't resemble it at all.
SideburnsOfDoom 2y ago
Will it be that fractal? it seems tenuous. Will it be a fractal? That would make sense.
pixelpoet 2y ago
I've spent a good portion of my life working on fractal software, trying basically every fractal rendering algorithm there is, reading everything I can about it, managing fractal art communities and organising meetups... so I'm obviously very pro-fractals biased, and even so, I'm curious what makes you conjecture so strongly that 3-body stability islands would have anything to do with the Mandelbrot set. Why that one in particular, over the vast infinitude of possible fractal shapes?

didgeoridoo 2y ago

Does anyone know if that visualization is representative or just an illustration? I’m struggling with how all these parameters (mass, speed, and relative direction for all three objects) could map to a fractal on a 2 dimensional plane.

rcxdude 2y ago
The study didn't plot the entire state space. It's simply a map of the end state of a limited space of parameters (specifically, the mass of the objects is fixed, two of them are in a particular orbit, and basically only the angle and timing of when the third object approaches is varied). The plot just shows which object gets thrown out of the system for each initial configuration.
- didgeoridoo 2y ago
  That makes a huge amount more sense. Thanks!

oolonthegreat 2y ago

AFAIK, these islands of regularity/stability are a common aspect of chaotic systems, say the logistic map[1]. [1]: https://en.m.wikipedia.org/wiki/Logistic_map https://en.m.wikipedia.org/wiki/Logistic_map

Rallen89 2y ago
Off topic buying haven't seen a bit of code in Wikipedia to run to see a visualisation before
ddejohn 2y ago
I may be mistaken but I seem to recall having also read on numerous occasions that ejection of one of the bodies is an inevitability in these types of systems. Given the limited state space of the simulation, I'm not sure I see what the big discovery is here. It's certainly a neat result to see it visualized though.

pkoird 2y ago

I never really understood the 3 body problem. Am I correct in saying that given three bodies that attract one another, there is currently no known way of predicting, at arbitrary time, where they'll end up at, even though it is possible to simulate to reasonable degree of accuracy for a limited time horizon? If that is the case, is this an example of what Stephen Wolfram regularly calls as computationally irreducible? Also it is so interesting to me that the jump from 2 to 3 in different domains herald such unfathomable complexity. 2SAT for instance is polynomially solved whereas 3SAT is NP Hard. If I were a competent mathematician, I'd definitely investigate the link between NP Hardness and Computationally irreducible chaotic process like 3Body problem.

nightowl_games 2y ago
Yes, I believe it is the same as Wolfram's "Computational Irreducability". I'll copy paste the question I emailed him about it: > In computers, floating point numbers have a limited precision, based on how many bits are used. Does the innate 'error' in floating point numbers have implications on your theory of 'Computational Irreducability'? Are the 'errors' introduced with each floating point operation is fundamentally computationally irreducable? Does this have a domino effect that causes basic classical mechanics to become computationally irreducible? Like the integration of velocity into position, it is a O(1) operation to compute the future position given a velocity and a starting point, but if a computer is rendering each individual frame, it will accumulate error in each frame, and the final frame may end up in a different position. Does this concept relate to the real world through things like the Planck length? Is space perhaps not purely continuous?
- ramesh31 2y ago
  >Is space perhaps not purely continuous? Pretty sure that's the entire basis of quantum mechanics
  - metalliqaz 2y ago
    space? or energy?

pradn 2y ago

The paper is at [1], and the summary chart is at [2]. The top chart shows which of the three bodies escapes the system given initial starting conditions. The bottom one makes it easier to see patterns by reducing noise. It uses the k-nearest-neighbors algorithm to find the dominant color near each pixel. More than the large pools of stability, what's interesting are the bands of stability - like electromagnetic fields. I have no idea how one codes this up in a computer, with fixed precision math and all. Numerical methods is a black box for me. [1]: https://doi.org/10.1051/0004-6361/202449862 https://doi.org/10.1051/0004-6361/202449862 [2]: https://www.aanda.org/articles/aa/full_html/2024/09/aa49862-24/F3.html https://www.aanda.org/articles/aa/full_html/2024/09/aa49862-...

jmpwat 2y ago

Rehydrate! Rehydrate!