The "CNS" is as far as I can tell just the use of a high order Taylor series together with multiple precision arithmetic for solving ODEs. This is a well known method for long term simulation of dynamical systems that has been around for decades. The authors basically imply that they invented this method and were the first to be able to study long term evolution of dynamical systems reliably because of it. Fair enough if they just weren't aware of previous work (which happens all the time), but it's an oversight that shouldn't have passed peer review.

That said, using this method to discover new periodic solutions of the three-body system is a very neat application, which deserves applause!

Are you sure that their numerical method isn't new? The abstract says that the author created it in 2009 and this paper further develops it to apply to the three body problem. If it already existed, can you say why it had not already been applied to the three body problem?

In fact the more general N-body problem is perhaps one of the most classical uses of high order Taylor methods, in particular for studying the long term stability of the solar system. Here is a paper from 1993: http://adsabs.harvard.edu/full/1993A%26A...272..687L. Rigorous error analysis of Taylor methods for ODEs goes all the way back to Moore's original work on interval arithmetic in the 1960s; in the early 2000s Makino and Berz developed so-called Taylor models which combine rigorous error bounds with accurate long-term propagation of changes due to perturbations in the initial values (see http://bt.pa.msu.edu/index_TaylorModels.htm); they used it to study the dynamics of the solar system among other things. R. Barrio and others have published several papers about reliable solution of ODEs using Taylor methods; leading to the development of the TIDES software. See several references listed on http://cody.unizar.es/tides.html. In particular the paper https://doi.org/10.1016/j.amc.2004.02.015 from 2005 investigates using variable order Taylor series with extended precision for the solutions of dynamical systems. The paper http://dx.doi.org/10.1155/2012/716024 from 2012 is specifically about obtaining periodic solutions of dynamical systems using a high order Taylor method with multiple precision arithmetic.

Actually, I made the first comment after I checked the second author's earlier paper https://arxiv.org/abs/1109.0130 where they essentially made the claim about inventing "CNS" as the first-ever reliable technique for long-term solutions of dynamical systems, without referencing any of the earlier work I mentioned above.

However, I just checked the actual text of this new article on the three-body problem (through sci-hub) and there the authors do cite the earlier work I mentioned above, giving proper attribution to others for the basic ideas behind what they call "CNS"! So all is actually well, and the peer review presumably did work (or the authors found out about the earlier work even before writing the new paper). It is just the press release that is misleading, as usual.

To answer your question, I'm not really familiar with the research on the N-body problem, so I can't say why or whether no one tried looking for periodic solutions in this way before. Perhaps no one actually thought of it, or they didn't try since they didn't expect to find anything, or they didn't have the computational resources, or they just couldn't figure out the details of how to do it (which the present authors did, and deserve credit for). Again, I was not trying to downplay the significance of this work, and finding new applications of existing methods (and making even tiny improvements along the way) is how science progresses. It also happens all the time that methods get rediscovered/reinvented independently.

The novel part is not the use of Taylor series for simulation. CNS stands for Clean Numerical Solution. The method of correcting the numerical error in the simulation is what is claimed to be novel.

From the CSN paper:

"the residual and round-off errors are verified and estimated carefully by means of different time-step Δt, different precision of data, and different order M of Taylor expansion....for the considered problem, the truncation and round-off errors of the CNS can be reduced even to the level of 10^−1244 and 10^−1000, respectively, so that the micro-level inherent physical uncertainty of the initial condition (in the level of 10^−60) of the H\'{e}non-Heiles system can be investigated accurately."

This is not a new idea. If you have a variable-order, variable-precision implementation, it's the most obvious way to estimate the error of a numerical solution.

I'm halfway through The Three Body Problem, a science fiction novel by Chinese author Cixin Liu. The apparent irrationality of three-body orbits is central to the story. So far, it's excellent, both as science fiction and as social commentary on contemporary China.

- many problems described are obvious in hindsight, but something you would never think about

Cons:

- Writing is uneven. Some parts you sail through, some parts make you physically cringe

- Many (too many) parts are "here's a scientific theory I learned about, let me give you a brief Wikipedia-style description. (Contrast with Blindsight where explanation of theories used is done in the appendix to the book, and is an interesting read on its own)

- No person in the history of this world has ever spoken like characters in the book. Dialogs are just people explaining things to each other

- Some things are introduced "just because". Otherwise the crises in the book couldn't be solved (e.g. it's assumed that the book talks about now and suddenly there are advanced nanomaterials and working hibernation)

All in all:

It's a very solid "hard" science fiction book which could be absolutely brilliant. But it's greatness is greatly reduced by the language and structure.

Liu escalates the arms race between races in the last third of the book so far that i was genuinely shocked and mildly unnerved at the implications of it. The trilogy really should have been called "The Dark Forest" because that is precisely what is described therein.

The trilogy is actually named "Remembrance of Earth's Past". That name is apt enough, though a bit bland. The Dark Forest is a pretty good concept though, and much more important to the story than the three body problem.

Yes, maybe, but superluminal communication could come from any number of places if that's true. Quantum entanglement has some very strong results, both experimental and mathematical that appear to exclude it from the running.

Like, I'd be more happy with wormholes or tachyons or "actually the universe is a simulation" as possible fictional sources of superluminal communication than entanglement, just because we really do know how entanglement behaves quite well by now and it's no good for FTL communication.

Special relativity is a fact. It is literally how the universe works. I don't know what else to say. There's no debate, it's a certainty that you cannot transmit information faster than light. Space and Time in Special Relativity is 240 pages, and you only really need to read like half of them.

Maybe gravity doesn't exist on the sun. Maybe electrons attract in the Sahara desert. This is what you're suggesting.

Authors are making a common mistake, and you're justifying it. The point of science fiction (at least hard scifi[0][1]) is to predict the future given some assumptions or premise like "we can control individual neurons"[2] or "there's life on a neutron star"[3] or "the moon exploded"[4] or "message from outer space"[5]. You get some outlandish sounding ideas that readers can consider seriously because the author genuinely thought them through. Ideas that don't respect relativity are useless. It's like writing fiction about Napoleon with walkie-talkies. It doesn't make sense, you could (probably) do much better than Napoleonic tactics with a walkie-talkie. A military tactic in space that relies on instant communication is a useless idea. Plus I just plain think it would've made a better story if they had to strategize around the speed of light.

(spoiler) to be fair, The Three Body Problem almost did it right, their ships travel at non-relativistic speeds, and the first 3 messages take 5 years to transmit. But I didn't bother reading past the first book after the quantum entanglement mistake on the last few pages.

Maybe I'm getting elitist. The hard/soft scifi thing is an old conversation. I understand that the social commentary is a big aspect of some books, just don't call them science fiction.

>Special relativity is a fact. It is literally how the universe works.

not exactly. SR is true only in the fixed Euclidean metric space. It is applicable to our Universe only as far as our Universe can be approximated by such a metric space. Which is really a very good approximation - almost like a "fact" - at relatively small (yet not extremely small) scales and at small mass/energy densities.

>There's no debate, it's a certainty that you cannot transmit information faster than light.

in the domain of applicability of SR. Ie. until, say, you do something severe to that metric space. Which author actually did, in some way, by folding/unfolding additional dimensions.

>Plus I just plain think it would've made a better story if they had to strategize around the speed of light.

reminded how USSR armed forces seriously strategized around cavalry attacks on the Eve of WWII, ie. the war which opened for USSR by lightning speed attack by German tanks and airforce.

The math behind GR and QM contradict each other. We have a large amount of empirical evidence for a speed limit for information travel, yes, but since we don't even know how the universe works (ie a mathematically consistent model, with few parameters, supported by experimental evidence), I don't think you can say "it's how the universe works". We might eventually find out that the information speed limit only holds at low energy scales for example.

Would you be happier with handwaved wormholes? I think you get the same effect but without so obviously violating the rules. (the practical requirements to build a wormhole and the difficulty in keeping it open aside, wormholes aren't completely excluded by general relativity, yeah?)

> Today, chaotic dynamics are widely regarded as the third great scientific revolution in physics in 20th century, comparable to relativity and quantum mechanics.

I wonder if that tells you something about the author's age. Chaos theory was the next big thing in the '80s and '90s - recall that Jeff Goldblum played a chaos theorist in Jurassic Park in 1993! It seems much less exciting today. I think it gave way to string theory, and now we have AI (yet again).

It seems like in the late 80s and early 90s this topic was very hot, then became a fad. You can find many books and papers that all have basically the same content (Lorenz/Henon attractors, logistic map, Mandelbrot set). I think it underwent a name change to "complexity science" during the 90s and 00s. I'm not in academia, but I do try to follow the topic because it is truly fascinating and IMO more accessible than quantum or relativity physics because many of the systems involved can be easily programmed/visualized with basic computers.

I've never heard such a thing, but if you think of "chaotic dynamics" as responsible for the dramatic improvement in weather prediction, it might start to make sense.

The site hosting the videos is down. Are there only 6 families (as in the caption at the top) or 600? Are all of the orbits discussed in the article in 2D or are some of them 3D? At the end it says the new CNS technique found only 243 new families, so how were the hundreds of other new ones found?

CNS from the article, " clean numerical simulation" enabled them to find 243 more than what the computational power was capable of finding with a lossy representation of periodicity. In other words, the strength of the machine using old-style algorithms for detection would have gotten 357+ solutions, and their use of CNS which cleans up the math a lot for the supercomputer helped them find an additional 243 that would have slipped through the cracks.

There are hints of that concern in TFA where they state the importance of accurate simulations over a long time. On the other hand, a simple two body system is very stable but will diverge in a simulation using a simple Euler integrator.

So here's an odd thing. I made a puzzle game for Android where I use Euler, and some of the puzzles require the divergance to work. It has caused a curious phenomenon where some people think the game is wrong, yet the game at no stage mentions what the rules of motion are.

I have seen people talk about the puzzles in terms of planets, gravity, massless bodies etc. when in-fact they are just circles that move towards or away from other circles. Theoretically people who can identifiy the Euler intergrator should be able to go "aha! I know what to do now" but so many get hung up on the idea that it is wrong.

The close encounters will cause issues even in integrators more sophisticated than Euler. One body of research used regularizing integrators to change variables when the bodies get close, to avoid numerical instabilities.

From a complete layman's point of view, I wonder if it's appropriate to think of a three-body system as somehow irreducible. In other words, maybe a "closed-form" representation of three-body motion can be defined in terms of a finite combination of stable periodic configurations. Anyway, just some musings.

So if the problem was first proposed by Newton in the 17th century, how did early investigators simulate the interactions between the bodies up until the electronic computer became available. There are no closed form solutions A.F.A.I.K. so they must have calculated the body's state by hand, I guess. How tedious.

My probably naive intuition says that if you had a system with two "suns" in the centre rotating around each other very closely, and one distant planet rotating about the centre of mass of the two suns, that would be stable surely?

Yes, that does work. In the literature this is called a "heirarchical system"

The very simplest version is called the Euler problem. Two fixed masses and a third moving in the "dipole field". All the solutions can be explicitly determined (although only in terms of e.g. Jacobi elliptic functions and other elliptics). There's a book "Integrable Systems in Celestial Mechanics" by Mathuna.

Right, but this dipole solution will neglect the perturbation of the third body on whatever is creating the dipole. Fair enough, because we expect the effect to be tiny.

But still you would have to unpack that dipole approximation to figure out if this perturbation will slowly change it in ways that do something significant in the long run.

as we know it is too late, trisolaris fleet is already under way, and the problem with the human science progress is so apparent that it even became a subject of the recent Big Bang Theory episode.

For those downvoting, the above comment is a reference to a terrific sci-fi novel, "The Three-Body Problem," which focuses on the same problem discussed in the article.

The "CNS" is as far as I can tell just the use of a high order Taylor series together with multiple precision arithmetic for solving ODEs. This is a well known method for long term simulation of dynamical systems that has been around for decades. The authors basically imply that they invented this method and were the first to be able to study long term evolution of dynamical systems reliably because of it. Fair enough if they just weren't aware of previous work (which happens all the time), but it's an oversight that shouldn't have passed peer review.

That said, using this method to discover new periodic solutions of the three-body system is a very neat application, which deserves applause!

Are you sure that their numerical method isn't new? The abstract says that the author created it in 2009 and this paper further develops it to apply to the three body problem. If it already existed, can you say why it had not already been applied to the three body problem?

In fact the more general N-body problem is perhaps one of the most classical uses of high order Taylor methods, in particular for studying the long term stability of the solar system. Here is a paper from 1993: http://adsabs.harvard.edu/full/1993A%26A...272..687L. Rigorous error analysis of Taylor methods for ODEs goes all the way back to Moore's original work on interval arithmetic in the 1960s; in the early 2000s Makino and Berz developed so-called Taylor models which combine rigorous error bounds with accurate long-term propagation of changes due to perturbations in the initial values (see http://bt.pa.msu.edu/index_TaylorModels.htm); they used it to study the dynamics of the solar system among other things. R. Barrio and others have published several papers about reliable solution of ODEs using Taylor methods; leading to the development of the TIDES software. See several references listed on http://cody.unizar.es/tides.html. In particular the paper https://doi.org/10.1016/j.amc.2004.02.015 from 2005 investigates using variable order Taylor series with extended precision for the solutions of dynamical systems. The paper http://dx.doi.org/10.1155/2012/716024 from 2012 is specifically about obtaining periodic solutions of dynamical systems using a high order Taylor method with multiple precision arithmetic.

Actually, I made the first comment after I checked the second author's earlier paper https://arxiv.org/abs/1109.0130 where they essentially made the claim about inventing "CNS" as the first-ever reliable technique for long-term solutions of dynamical systems, without referencing any of the earlier work I mentioned above.

However, I just checked the actual text of this new article on the three-body problem (through sci-hub) and there the authors

docite the earlier work I mentioned above, giving proper attribution to others for the basic ideas behind what they call "CNS"! So all is actually well, and the peer review presumably did work (or the authors found out about the earlier work even before writing the new paper). It is just the press release that is misleading, as usual.To answer your question, I'm not really familiar with the research on the N-body problem, so I can't say why or whether no one tried looking for periodic solutions in this way before. Perhaps no one actually thought of it, or they didn't try since they didn't expect to find anything, or they didn't have the computational resources, or they just couldn't figure out the details of how to do it (which the present authors did, and deserve credit for). Again, I was not trying to downplay the significance of this work, and finding new applications of existing methods (and making even tiny improvements along the way) is how science progresses. It also happens all the time that methods get rediscovered/reinvented independently.

The novel part is not the use of Taylor series for simulation. CNS stands for Clean Numerical Solution. The method of correcting the numerical error in the simulation is what is claimed to be novel.

From the CSN paper:

"the residual and round-off errors are verified and estimated carefully by means of different time-step Δt, different precision of data, and different order M of Taylor expansion....for the considered problem, the truncation and round-off errors of the CNS can be reduced even to the level of 10^−1244 and 10^−1000, respectively, so that the micro-level inherent physical uncertainty of the initial condition (in the level of 10^−60) of the H\'{e}non-Heiles system can be investigated accurately."

This is not a new idea. If you have a variable-order, variable-precision implementation, it's the most obvious way to estimate the error of a numerical solution.

I'm halfway through The Three Body Problem, a science fiction novel by Chinese author Cixin Liu. The apparent irrationality of three-body orbits is central to the story. So far, it's excellent, both as science fiction and as social commentary on contemporary China.

It's a brilliant trilogy in a shitty packaging.

From a personal recommendation elsewhere:

Pros:

- the premise is good

- many of the plot twists are good or brilliant

- many problems described are obvious in hindsight, but something you would never think about

Cons:

- Writing is uneven. Some parts you sail through, some parts make you physically cringe

- Many (too many) parts are "here's a scientific theory I learned about, let me give you a brief Wikipedia-style description. (Contrast with Blindsight where explanation of theories used is done in the appendix to the book, and is an interesting read on its own)

- No person in the history of this world has ever spoken like characters in the book. Dialogs are just people explaining things to each other

- Some things are introduced "just because". Otherwise the crises in the book couldn't be solved (e.g. it's assumed that the book talks about now

andsuddenly there are advanced nanomaterials and working hibernation)All in all:

It's a very solid "hard" science fiction book which could be absolutely brilliant. But it's greatness is greatly reduced by the language and structure.

>

Many (too many) parts are "here's a scientific theory I learned about, let me give you a brief Wikipedia-style description.So? Doesn't sound that different from 90% of classic sci-fi, especially Asimov.

>

- No person in the history of this world has ever spoken like characters in the book. Dialogs are just people explaining things to each otherDitto.

> So? Doesn't sound that different from 90% of classic sci-fi, especially Asimov.

That doesn’t make it good writing.

No, but it means it can still be good sci-fi.

Sci-fi was never about the writing, in the way hi-brow literature was.

Heck, it started from pulp magazines aimed at teenagers.

amazing series! book 2 is probably the most mindblowing sf novel i've read since brin's "startide rising"

Have you read Nexus yet?

That would be more literally mind blowing :-) but I can heartily recommend it.

yes, and i liked it, but it wasn't nearly as stunning as the three body trilogy was.

I'd second this recommendation: Nexus is one of the most interesting and entertaining sci-fi novels I've read in recent years.

I'd heard the sequel was not so good—anyone have an opinion?

I loved the whole trilogy, but that may very well be because I just love the concepts in the story.

And I'm halfway through

Death's End(the third volume in the series). I didn't think this was possible but I can tell you that it gets much.much what? Don't leave me hanging.

'Bigger' is the word I would use. The scope of the story is incredible

I recently described the scope as an exponential function, which is really the only acceptable way of doing justice to the book.

Yeah, that is one way of putting it.

Liu escalates the arms race between races in the last third of the book so far that i was genuinely shocked and mildly unnerved at the implications of it. The trilogy really should have been called "The Dark Forest" because that is precisely what is described therein.

The trilogy is actually named "Remembrance of Earth's Past". That name is apt enough, though a bit bland. The Dark Forest is a pretty good concept though, and much more important to the story than the three body problem.

I've read it a couple of months ago. Interesting idea but terrible book overall. Very naïve. Will definitely not read the second part.

Could you elaborate? I usually read positive stuff about this book, so I'm very curious to hear diverging views.

Part two and part three are both exceedingly more brilliant. Very much advise you to read them.

I, for one, found it a masterpiece.

Books 2/3 are waaaay better! You’re missing out

One of the best hard sci-fi series I've ever read, hands down.

What do you prefer?

(minor spoiler) in my opinion, any book that violates the theory of relativity is not science fiction, it's a fantasy novel set in "space".

You can't transmit information faster than light using quantum entanglement.

https://en.wikipedia.org/wiki/Faster-than-light#Quantum_mech...

Maybe the laws of our math and physics are not invariant to space and time :)

Yes, maybe, but superluminal communication could come from any number of places if that's true. Quantum entanglement has some very strong results, both experimental and mathematical that appear to exclude it from the running.

Like, I'd be more happy with wormholes or tachyons or "actually the universe is a simulation" as possible fictional sources of superluminal communication than entanglement, just because we really do know how entanglement behaves quite well by now and it's no good for FTL communication.

Special relativity is a fact. It is literally how the universe works. I don't know what else to say. There's no debate, it's a certainty that you cannot transmit information faster than light. Space and Time in Special Relativity is 240 pages, and you only really need to read like half of them.

Maybe gravity doesn't exist on the sun. Maybe electrons attract in the Sahara desert. This is what you're suggesting.

Authors are making a common mistake, and you're justifying it. The point of science fiction (at least hard scifi[0][1]) is to predict the future given some assumptions or premise like "we can control individual neurons"[2] or "there's life on a neutron star"[3] or "the moon exploded"[4] or "message from outer space"[5]. You get some outlandish sounding ideas that readers can consider seriously because the author genuinely thought them through. Ideas that don't respect relativity are useless. It's like writing fiction about Napoleon with walkie-talkies. It doesn't make sense, you could (probably) do much better than Napoleonic tactics with a walkie-talkie. A military tactic in space that relies on instant communication is a useless idea. Plus I just plain think it would've made a better story if they had to strategize around the speed of light.

(spoiler) to be fair, The Three Body Problem almost did it right, their ships travel at non-relativistic speeds, and the first 3 messages take 5 years to transmit. But I didn't bother reading past the first book after the quantum entanglement mistake on the last few pages.

Maybe I'm getting elitist. The hard/soft scifi thing is an old conversation. I understand that the social commentary is a big aspect of some books, just don't call them science fiction.

[0] https://news.ycombinator.com/item?id=15462327

[1] https://www.google.ca/search?q=three+body+problem+"hard+scie...

[2] https://en.wikipedia.org/wiki/The_Nexus_Trilogy

[3] https://en.wikipedia.org/wiki/Dragon%27s_Egg

[4] https://en.wikipedia.org/wiki/Seveneves

[5] https://en.wikipedia.org/wiki/His_Master%27s_Voice_(novel)

https://books.google.ca/books/about/Space_and_Time_in_Specia...

https://news.ycombinator.com/item?id=8937197

https://news.ycombinator.com/item?id=7888847

https://news.ycombinator.com/item?id=4588938

>Special relativity is a fact. It is literally how the universe works.

not exactly. SR is true only in the fixed Euclidean metric space. It is applicable to our Universe only as far as our Universe can be approximated by such a metric space. Which is really a very good approximation - almost like a "fact" - at relatively small (yet not extremely small) scales and at small mass/energy densities.

>There's no debate, it's a certainty that you cannot transmit information faster than light.

in the domain of applicability of SR. Ie. until, say, you do something severe to that metric space. Which author actually did, in some way, by folding/unfolding additional dimensions.

>Plus I just plain think it would've made a better story if they had to strategize around the speed of light.

reminded how USSR armed forces seriously strategized around cavalry attacks on the Eve of WWII, ie. the war which opened for USSR by lightning speed attack by German tanks and airforce.

> Special relativity is a fact.

The math behind GR and QM contradict each other. We have a large amount of empirical evidence for a speed limit for information travel, yes, but since we don't even know how the universe works (ie a mathematically consistent model, with few parameters, supported by experimental evidence), I don't think you can say "it's how the universe works". We might eventually find out that the information speed limit only holds at low energy scales for example.

Would you be happier with handwaved wormholes? I think you get the same effect but without so obviously violating the rules. (the practical requirements to build a wormhole and the difficulty in keeping it open aside, wormholes aren't completely excluded by general relativity, yeah?)

> Today, chaotic dynamics are widely regarded as the third great scientific revolution in physics in 20th century, comparable to relativity and quantum mechanics.

Really??

I wonder if that tells you something about the author's age. Chaos theory was the next big thing in the '80s and '90s - recall that Jeff Goldblum played a chaos theorist in Jurassic Park in 1993! It seems much less exciting today. I think it gave way to string theory, and now we have AI (yet again).

String theory is unproven, and AI is a technology, not a scientific theory.

Here is an article that shares the opinion and gives an overview of key ideas.

https://arxiv.org/abs/1306.5777

It seems like in the late 80s and early 90s this topic was very hot, then became a fad. You can find many books and papers that all have basically the same content (Lorenz/Henon attractors, logistic map, Mandelbrot set). I think it underwent a name change to "complexity science" during the 90s and 00s. I'm not in academia, but I do try to follow the topic because it is truly fascinating and IMO more accessible than quantum or relativity physics because many of the systems involved can be easily programmed/visualized with basic computers.

I've never heard such a thing, but if you think of "chaotic dynamics" as responsible for the dramatic improvement in weather prediction, it might start to make sense.

The result videos from the authors is located at http://numericaltank.sjtu.edu.cn/three-body/three-body.htm

Funny thing is, it can not be opened, because of the 19th national congress. All .edu.cn second-level domain were shutdown for "security reasons"

Huh. This is the first time I can access a website from China, but not when using a VPN. Here's a random GIF of orbital dynamics for you: https://imgur.com/kzPcL4h (Mirror of http://numericaltank.sjtu.edu.cn/three-body/unequal-mass-dat...)

The site hosting the videos is down. Are there only 6 families (as in the caption at the top) or 600? Are all of the orbits discussed in the article in 2D or are some of them 3D? At the end it says the new CNS technique found only 243 new families, so how were the hundreds of other new ones found?

CNS from the article, " clean numerical simulation" enabled them to find 243 more than what the computational power was capable of finding with a lossy representation of periodicity. In other words, the strength of the machine using old-style algorithms for detection would have gotten 357+ solutions, and their use of CNS which cleans up the math a lot for the supercomputer helped them find an additional 243 that would have slipped through the cracks.

The pre-print from arxiv: https://arxiv.org/abs/1705.00527v4

Are any of them dynamically stable?

Yeah, that's what I'm curious about. Would all of them deviate if they were breathed on wrong? Could any of them occur in nature?

There are hints of that concern in TFA where they state the importance of accurate simulations over a long time. On the other hand, a simple two body system is very stable but will diverge in a simulation using a simple Euler integrator.

So here's an odd thing. I made a puzzle game for Android where I use Euler, and some of the puzzles require the divergance to work. It has caused a curious phenomenon where some people think the game is wrong, yet the game at no stage mentions what the rules of motion are.

I have seen people talk about the puzzles in terms of planets, gravity, massless bodies etc. when in-fact they are just circles that move towards or away from other circles. Theoretically people who can identifiy the Euler intergrator should be able to go "aha! I know what to do now" but so many get hung up on the idea that it is wrong.

Are you allowed to tell us the name of the game, so we can have a look?

The close encounters will cause issues even in integrators more sophisticated than Euler. One body of research used regularizing integrators to change variables when the bodies get close, to avoid numerical instabilities.

This integrator is used in my three body app, to remove the issues due to close encounters. An old blog post of mine has the details: http://nbodyphysics.com/blog/2015/12/08/threebody-2-0/

From a complete layman's point of view, I wonder if it's appropriate to think of a three-body system as somehow irreducible. In other words, maybe a "closed-form" representation of three-body motion can be defined in terms of a finite combination of stable periodic configurations. Anyway, just some musings.

Irreducible is a good way to describe the equally-balanced constraint in a 3-body situation.

Is there an upper limit on how many different periodic orbitals a 3-body system may have?

Generally if you could prove that there was, or that their wasn't, you would probably win the Fields Medal at least.

So if the problem was first proposed by Newton in the 17th century, how did early investigators simulate the interactions between the bodies up until the electronic computer became available. There are no closed form solutions A.F.A.I.K. so they must have calculated the body's state by hand, I guess. How tedious.

With some symmetry you can find a couple of solutions, which is what Laplace and Lagrange did. e.g. http://www.phys.lsu.edu/faculty/gonzalez/Teaching/Phys7221/T...

The scholarpedia article is a good meta-reference:http://www.scholarpedia.org/article/Three_body_problem

My probably naive intuition says that if you had a system with two "suns" in the centre rotating around each other very closely, and one distant planet rotating about the centre of mass of the two suns, that would be stable surely?

Yes, that does work. In the literature this is called a "heirarchical system"

The very simplest version is called the Euler problem. Two fixed masses and a third moving in the "dipole field". All the solutions can be explicitly determined (although only in terms of e.g. Jacobi elliptic functions and other elliptics). There's a book "Integrable Systems in Celestial Mechanics" by Mathuna.

I recently added the Euler problem to my iOS app ThreeBody: https://appadvice.com/app/threebody-lite/951920756

Some day I'll get around to adding these 600 solutions...

Right, but this dipole solution will neglect the perturbation of the third body on whatever is creating the dipole. Fair enough, because we expect the effect to be tiny.

But still you would have to unpack that dipole approximation to figure out if this perturbation will slowly change it in ways that do something significant in the long run.

It was probably an early form of spirograph that helped solve some of the trajectories.

You can find a bunch of n-body solutions, some of which like those in this article are pretty amazing, on Cris Moore's gallery page http://tuvalu.santafe.edu/~moore/gallery.html

as we know it is too late, trisolaris fleet is already under way, and the problem with the human science progress is so apparent that it even became a subject of the recent Big Bang Theory episode.

For those downvoting, the above comment is a reference to a terrific sci-fi novel, "The Three-Body Problem," which focuses on the same problem discussed in the article.

fortunately, we live in a dark forest, and need only threaten to shine a light to make the invasion fleet pointless...

Yeah, but that doesn't always turn out so well :(