That Pythagorean one with the ejection is cool. I wonder whether you could use it to accelerate a spacecraft.

Cool, are any of these particularly stable? Like, is there any choreography that could plausibly survive in practice?

> I'm not sure what to do with the information

Get inspired and spend a rest of your life looking for new 3bp solutions.

Something cool and interesting for its own sake. A rare find.

The three-body problem only states the problem to solve, it doesn’t itself state anything about the existence or non-existence of solutions. It has been proven that there is no general closed-form solution. And there are obvious solutions for trivial special cases, such as three equal masses in an equilateral triangle rotating around each other.

Further reading: https://en.wikipedia.org/wiki/Three-body_problem#Solutions

There is no closed-form solution for finding the roots of >4th degree polynomials in general, but that doesn’t preclude many families of >4th degree polynomials from having closed-form solutions. As a trivial example, x^5 - 1. The exact same thing with the three-body problem.

In the case of a planet going around the sun we know that the planet travels in an ellipse, more generally a conic section including interstellar comets. Orbital periods and everything else can be computed straightforwardly with formulas.

In the three body problem you can always do a numerical integration (e.g. simulation) and this is valid for a certain amount of time but will not be valid forever because of: (1) chaotic motion which amplifies small errors exponentially over time and (2) celestial mechanics is symplectic which means it conserves certain geometric properties and most integrators are not symplectic and must have different long term dynamics. There are symplectic integrators but they don’t work as well overall as ordinary integrators.

We do not know, for instance, if the solar system is stable. In the short term the planets seem to be basically doing their own thing in their orbits with just minor perturbations. We know the orbits vary a bit over millions of years. We aren’t sure which side of the sun Pluto will be on in 30 million years. It’s very believable that the planets are doing the same thing in 4 billion years but we can’t rule out that the orbits could change in a big way or one could get ejected.

The general case always has a solution. At least until the point where two of the three bodies meet (which is a singularity). We can approximate that solution numerically.

The problem is that the solutions very strongly tend to be chaotic. Meaning that small differences in initial conditions, tend to grow exponentially with time. Which means that if you measure everything to 3 digits of precision, in finite time it will stop looking like the actual solution. Every additional digit of precision adds a similar finite time to how long the approximation is good for.

So when finally found, say, the 1953 BC conjunction described in https://en.wikipedia.org/wiki/Conjunction_%28astronomy%29?#N... - that was a very good stress test for our estimated planetary data. Because surprisingly small errors in modern data would have kept that conjunction from happening.

Yes. It's kind of like the halting problem: You cannot write a general computer program that will analyze the source code of any random other computer program and tell you if it will halt.

You can write a program that will analyze the code of a few specific other programs and tell you if they will halt. You just can't do it in general.

The 3-body problem is like that. Except it's much harder to find stable 3-body problems than computer programs that are predictable.

I'm pretty sure there is always a unique solution to the equations of motions (safe for some pathological edge cases perhaps). Classical mechanics is deterministic, after all. But for more than two bodies, there is in general no solution in closed form, and it's often chaotic, so not even computeable for arbitrary time frames.

The "about" info states that all of these are computed numerically.

There is a general solution to the three body problem due to Sundman, in the form of an infinite series. Unfortunately it converges so slowly that it's useless in practice.

Being unable to state the solution of a problem in closed form doesn't prevent solutions from existing.

That's exactly the case

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A few that I looked at are annotated, "It is linearly stable (largest Floquet multiplier 1.000), so a small nudge only makes it wobble." Doesn't define "small nudge", alas...

further research indicates some do exist:

- equilateral triangle is the simple case

- some figure eight configurations

- some configurations that act like two nested binary systems

there might be others. fascinating stuff.

Really cool project! On a personal note, thanks for the reference! I'm one of the authors of the 2013 paper you cited, and it's awesome seeing how many new solutions have been found since then.

I have a quick question: how are you currently checking if a new candidate is genuinely novel rather than a rotated, time-shifted, or rescaled duplicate of an existing solution?

If you are not already computing it, topological classification (mapping orbits to free group words) would be a great way to quickly prune duplicates before heavy numerical checks, and it's also really useful metadata to display in the atlas itself. Also, showing shape sphere projections of the orbits would be a nice visual addition.

How did you end up choosing the color scheme for this site?

The Trisolarians are supposedly at Alpha Centauri, a trinary star that isn't actually that chaotic. It has two sun-like stars orbiting each other quite close, and then a red dwarf, Proxima Centauri, orbiting that pair really really far away.

Given all the illogical insanity they are doing (like, trying to fight an interstellar war instead of just the simple stuff like moving to habitats, improving their bodies or even so,me stellar lifting) I don't think the will listen. ;-)

While I love the books as much as everyone else, the pedant in me has to point out that the Trisolareans have a four body problem at hand (3 suns and their planet), not a three body problem.

Any set of points (w. velocities) along the trajectories could be the "starting positions".

If you go to the individual solutions, the text description tells you if it's stable. There's also a slider that allows you to perturb the orbit so you can see for yourself when you perturb it.

I think that's what "STABLE ONLY" clickable text filters by.

They’re not stable except at L4 and L5, and they all assume oke body to be massless. Arguably they are 2-body orbits for that reason. Not sure but suspect that this atlas contains non-massless bodies.

Great suggestion, I added an option to have it in full screen mode that prevents sleep - click "Open Observatory" to enter full screen mode

Deterministic chaos is cool. But this is even more special in a sense: for a problem where random initial conditions are almost always chaotic, this is a catalog of periodic orbits—these are all non-chaotic.

It would be cool to pair this with a numerical simulator that shows what happens when you perturb any of them.

EDIT: oh, it already does this, thanks other comments

Yes. Because 3 points are coplanar, so every "3D problem" with 3 objects can be turned into a 2D problem on the correspondent plane

(of course in real life your plane would keep changing, and probably some other complicated math I can't think right now)

https://numericaltank.sjtu.edu.cn/three-body/three-body.htm shows that there are three dimensional solutions.

We have no observed examples in nature of three body equilibrium. But then again, all places we have looked are either influenced by the chaotic orbits around them of the Solar System, our surrounding galaxy, or nearby galaxies in a cluster.

There aren't a lot of orbiting three bodies without external gravitational influences disturbing them.

Wild Gears for the modern physical version.

There's also Spirograph Simulator (2014) https://news.ycombinator.com/item?id=13256222 from... oh, a decade ago (I feel old). It still works. https://inspiral-web.nathanfriend.com

It has a list of data sources here: https://www.threebodyorbits.com/about

This whole thing does have a strong "GPT-6 Astra look" with the big text on the left style. But it is indeed very cool.

I find the freefall ones to be... more wild.

https://www.threebodyorbits.com/orbit/freefall_part2_f2310_1... (change to a Short trail without a full trace)