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Mathematicians find way to put 7 cylinders in contact without using their ends (sciencenews.org)
265 points by ColinWright on April 5, 2014 | hide | past | favorite | 84 comments

It has been known for a long time that one can arrange 7 cylinders to be mutually touching. That was written about by Martin Gardner decades ago, and was set as a puzzle.

The result had the cylinders touching at the end of one with the length of the other, so the question arose, can one arrange to have seven cylinders all mutually touching, without using the ends. The easiest way to say this is to ask for seven infinitely long cylinders mutually touching.

This has only recently been settled, hence this paper. It's believed impossible to arrange eight identical infinitely long cylinders to be mutually touching. I suspect the result is in fact known, but I haven't searched diligently for it.

There is an associated puzzle that uses cylinders that are very short - think coins. How many coins can you arrange to be mutually touching?

Consider that a puzzle. I can do 5. If you can do more, there's a mathematical paper in it for you, should you care.

I linked to another related puzzle elsewhere in this discussion: there appears to be a solution for 9 cylinders of infinite length but different radii. It's not clear what's known about the case of 10 cylinders.

So there are known solutions for 5 coins, 7 identical infinite cylinders, 7 identical finite cylinders (maybe more), and 9 different infinite cylinders.

Is there an example of a similiar non-trivial problem for which there is an inexistance proof? I'm not a mathematician so I have a hard time picturing a way of proving this sort of thing when the number of objects is "tricky" (not too high, not too low), apart from simply showing a counterexample.

How about the fact that there are no more than 5 platonic solids [1]? (See the proof in the article)

1: http://en.wikipedia.org/wiki/Platonic_solid

Question I've not found the answer to:

    It's possible to have 7 arbitrarily long cylinders
    mutually touching.  Currently it's not possible to
    have more than 5 coins (which are short cylinders)
    mutually touching.  As the cylinder's aspect ratio
    decreases, where are the thresholds: 7 -> 6 -> 5 ?

This sort of reminds me of: http://en.wikipedia.org/wiki/G%C3%B6mb%C3%B6c

Because it is a 3D object that was found using mathematics. Any other examples?

I think there are lots of new objects discovered in higher dimensions, but I like when there is something you can actually build and see. I also like how it appears to be very asymmetrical.

The Klein bottle is sort of like that, only it's supposed to be in 4 dimensions, but the 3d version is still pretty cool and quite famous!


Never heard about the gomboc before, thanks!

Clifford Stoll of Cuckoo's Egg fame has a side business making glass Klein bottles. He lives near me, I sometimes see him at the post office shipping out a bunch of them.

His site has a cool domain name: http://www.kleinbottle.com

Answering my own question, here is a great one from an artist that I love: http://www.bathsheba.com/math/gyroid/

This is going to sound like trolling, but it's not - I'm honestly curious.

Why is this important? Is it just cool, or is there some real world application? Was someone paying for this research for some reason, or was it just a mathematician's hobby?

EDIT: For the record, I don't have any problem with "just cool" research. I do that kind of research often (albeit, not as smart), and totally understand the value in it. Just wondering if this had an application immediately.

There are many puzzles, games, and obsessions that require the creation of new mathematics to understand properly, and then which eventually prove to be useful.

Sometimes solving these sorts of problems requires pushing the boundaries of existing techniques, sometimes the techniques required are completely new and are invented specifically for the question at hand.

I'm not sure about this specific instance with regards payment, but sometimes people are paid to play with these sorts of things simply because you don't know what will come from it.

People constantly, constantly ask this question - "What's the point of pure research?" The question has been discussed many, many times over, and there are eloquent and robust defences for pure research to be found in many places. There's no way I could do better than some of the things already out there, so I won't try. However, many of the things you use today and take for granted for first created by people "playing," sometimes in their spare time, sometimes in time paid for by companies (and in some cases individuals) who were willing to let people play and see what came of it.

If we knew what we were doing, it wouldn't be called "Research."

My high school calculus teacher told me that even imaginary numbers (those infinitely useful things in EE) were originally createdas a pure mathematical tool with zero practical application.

He liked to rail against those "dirtyfilthyphysicists" stealing pure mathematical concepts and finding applications for it.

Sounds like G.H.Hardy

   "Nothing I have ever done is of the slightest practical use"

This from the man who came up with the Hardy-Weinberg Equation[1], an extremely practical method of population analysis.


When asked about this kind of topic I prefer to answer with the following joke:

'A mathematician, native Texan, once was asked in his class: "What is mathematics good for?" He replied: "This question makes me sick. Like when you show somebody the Grand Canyon for the first time, and he asks you `What's is good for?' What would you do? Why, you would kick the guy off the cliff".'

(Source: http://www.math.utah.edu/~cherk/mathjokes.html)

I'm not sure what you mean by "real world problems". Most mathematicians do mathematics because they find it beautiful, rewarding, fun, and for various other reasons that have little to do with creating things in the physical world. Mathematics is extremely interconnected and some minor discovery in a certain field can later become extremely important in some seemingly unrelated field, similarly - real world applications often arise unexpectedly out of things which at first seem completely theoretical (famous example: public key cryptography), and we cannot foresee in advanced what will lead to what.

Timothy Gowers - 1998 Fields medalist - gave a lecture in 2000 titled "The Importance of Mathematics". I advice everyone to watch/read it:

Video: https://www.youtube.com/view_play_list?p=3641A12A6ADDB4B1

PDF: https://www.dpmms.cam.ac.uk/~wtg10/importance.pdf

Edit: more philosophically, this question always seems puzzling to me, because we can always ask: whats so important about "real world applications" anyway? We can start from any activity and ask "why is it important?", and then ask the same about the answer, etc'. This chain of "why"s eventually comes to an end: we can still ask the question, but there is no meaningful answer other than the realization that the things which are important are the things people find important - whether it is to get faster from point A to point B, have a baby, go to a party, or to solve mathematical problems. Something isn't more important because it's in the "real world".

I think many "real world applications" tend to directly or indirectly support our biological goals of individual and/or species survival and reproduction.

Of course, the same could be said of basic research, but it is further removed and so requires more forward thinking to appreciate.

I don't see how this is true. If the question is "what are the real world applications of X in mathematics?", most people would be satisfied by any absurd answer as long as it's "in the real world", regardless of what biological goals, if any, it supports.

It could be that a certain mathematical research allowed us to develop sturdier pencils, and suddenly everybody is satisfied that we are not wasting our money because now it has "real world applications".

The human drive to discover and invent new things in math is fundamentally no different than the human drive for doing the same in the fields of technology, arts, etc'. But we perceive anything touchable as good, and anything imaginary as useless, because that's the kind of culture we live in: Making a slightly faster smartphone so that the activity of swiping a finger on a capacitive touch screen to play candy crush now takes 0.7 seconds instead of 0.8 doesn't need any justification, but thinking about math does.

About "just cool" thing: that discussion has been done a zillion times. On the one hand, it's just one of those funny problems: easy to describe, but far from simple to solve. On the other hand, I would say that mathematicians have earned the right to work on such silly problems, given that, if hey hadn't worked on similar silly problems before, our technology would be quite a bit less advanced. Half the world is built on the hobbies of mathematicians.

What surprised me is that http://arxiv.org/abs/1312.6207 (which gives solutions with 8 and 9 cylinders, if one let's go of the constraint that all cylinders must be of equal radius) claims a connection with physics.

(BTW, I found that link via http://www.mathpuzzle.com. Infrequently updated, but never has uninteresting new items (at least, to my taste))

If you ask some professional mathematicians, individually, why it is they're researching what they're researching, I think the most common answer will be "because I think it's interesting". There are areas of math without direct application to the real world. And so what? The idea that we should only do that which has some immediate gain is intellectually toxic. It stifles curiosity.

The techniques involved are both generic and highly non-trivial. So the paper is very nice in that it gives a good work-out to the methods, and the reader can get to see what kind of problem would yield to those methods. For example it appears that similar kinds of "objects touching each other" problems could be attacked. But yeah, it's basically a monster polynomial that they solve, and that is quite a common thing to want to do.

On the other hand, one might see a paper that expounds a technique, but how do you know if that technique has got any teeth unless you can use it to eat a difficult problem?

So yeah, I think the specific problem is not quite the point of this research.

Exploring geometric puzzles made Professor Ernő Rubik the richest and most famous man in communist Hungary [1].

At least that's a concrete answer you can give to the "man in street" who's going to scoff at explanations about expanding mathematics. But money and fame, well, everybody understands that if it makes money or makes you famous it's justified.

[1] http://www.nytimes.com/1986/08/03/business/hungarian-million...

>"everybody understands that if it makes money or makes you famous it's justified" //

I'm assuming that was tongue-in-cheek. Whilst many will accept the answer "to be famous" or "to make a profit" those are not justified motivations to others - particular not WRT research funding.

I believe this result has important applications in cylindrical stack encryption; it's very difficult to produce a solution, but very easy to verify a given solution.

You jest, but your comment made me think of what an impressive Enigma machine [1] such an arrangement of rotors could make. The tubes are slipping in contact, so it'd have to be electrical linkage rather than mechanical to conduct the flow of the data.

[1] http://en.wikipedia.org/wiki/Enigma_machine

Brb, I'm gonna create CylinderCoin.

I'd buy some. Would be awesome if each (coin) solution was rendered with a raytracer.

I came in wondering the same thing. Why is this newsworthy? Every one of the replies to your comment is either a joke or an obnoxious lecture about "why maths can just be for the sake of it [and might be useful later]" like you're some sort of recently-thawed caveman. People can't just say "no, no actual applications yet, it's just cool," or, "I don't know." Ugh. Whole thread made me cringe.

I think this would be an interesting sculpture for a campus math building. I can imagine John Raymond Henry doing it. If Claes Oldenburg made them into gigantic cigarettes, there's no telling what would happen ;-)

Great design for a no-gravity space station.

Easy to go everywhere from everywhere.

I was going to make a vaguely snide comment about difficulty in routing and managing services (air, water, power, data, &c.), but the more I look at the model, the less firmly I'm convinced that there is an intractable topological problem involved.

I'm still going to go with calling it the Villa Straylight, though.

You could dedicate a cylinder to utilities, and it will connect directly to all the others.

That was more or less what I had in mind. On the other hand, it might be more cost-efficient in general to design the cylinders to connect endwise and form a hexagon; it'd increase travel time for the station's inhabitants, but reduce the need for unique construction tooling -- instead of seven types of cylinders, each of whose interconnections are placed differently from those of all the others, you build one type of cylinder and one type of connector, then fit out each unit according to the purpose it serves in a given station design. You can also enable more complex topologies simply by developing more complex connectors, whereas what I shall please myself by calling the "eccentric" design under discussion doesn't scale nearly as well.

Or you could forego connectors and have them all intersect slightly. Once in space, pop out the panels where the intersections will happen.

That scales better than the eccentric design, but not as well as separate connectors.

Suppose, for example, you want to be able to assemble, from the same components, a flat-hexagon station, and an "asterisk" station like what you'd get if you drew radii from each vertex of a hexagon and then took the hexagon away -- a configuration which might be very useful, for example, as a "transfer point"; if the inter-segment connectors are adaptable to whatever standard governs spacecraft airlocks, then you can dock at least six spacecraft at once and interchange cargo among any or all of them. If you add a seventh "socket" to your central connector, normal to the plane of the other six, then you can tie your "transfer point" to a larger station, too, and assemble a larger structure, as for example might be very useful at a nearby Lagrange point, as a way station for ships inbound to and outbound from the Earth-Moon system.

(On the other hand, perhaps I simply spent too much time playing with Tinkertoy as a child.)

I'd love to see a jungle gym in this shape. I would definitely climb it!

I did: https://www.flickr.com/photos/jzwinck/252678921/in/set-72157...

It was great: whether by design or luck, the poles you had to climb got more and more horizontal as you got higher from the ground, then they became inverted in a way that made the topmost part easy and safer. The bottom then was a test: if you failed you had not far to fall, if you succeeded you would likely be ok to the top.

Was was thinking more of tubes with ladders inside. But this one looks great too!

Here's the original solution - http://www.mathpuzzle.com/7cylinders.gif

Ask HN: Is there an explanation somewhere of this "certification" that a layperson (college math and physics major, and self taught programmer) could understand?

The article is quite readable. I'm a layperson too, but here's what I made of the methods provided:

1- The first method finds a constant that guarantees convergence of newtons method to the solution; they then iterate it a few times and verify the result is almost stationary.

2- They use a set with parameter r that is guaranteed to contain a solution if it's contained in a ball of radius r; they then show that a given small ball around the candidate solution contains this set (they were able to do that because this set gets smaller faster than the radius).

So, given that this involves rounding errors in solutions to equations it is only an approximate solution. A solution with fuzz. Is it possible to prove that each point of "contact" is exactly coincident? Or to prove that exact coincidence is not possible. There seems to be room for deeper work on this problem.

The initial space for the solution was found with techniques that had the possibility of rounding errors, and then exact proofs were found. This is a standard technique, one that I used in my PhD. So yes, it is possible to prove that each point of "contact" is exactly coincident, and if you read the paper carefully you'll see that that's what the authors did.

I was wondering this as well. Perhaps this is an example of a new, computer-assisted paradigm of math, where after extensive computation, a proposition is accepted as true if the probability of its being false is judged to be sufficiently small. I don't have the exact reference, but there was an article in some AMS publication about this in a few years ago.

I believe that there was a mathematical proof published along with the news.

Manufacturing errors? I can't tell if this is plastic, but if it is, surely a machinist can do better with metal.

They're just saying that building it is not a proof - the computer simulation is more exact than the model, and even the simulation is subject to rounding errors in what is considered "touching".

Sounds like they used the computer to search for a proposed solution that _could_ have been a false positive due to rounding error, then verified it through exact mathematical methods.

By "errors" they mean "the minute imperfections created by molding, machining, carving, etc." The rounding errors in the computer model would produce smaller physical artifacts than the manufacturing method would.

Surely mathematicians would use rationals[1] rather than floating point numbers, thus eliminating rounding errors at the expense of performance?

[1] https://gmplib.org/

It is not a matter of rational numbers, since the solution is likely to be an irrational number.

The paper [1] describes a system of 20 polynomial equations with 20 variables (Equations 10--12) and solving them is no trivial task. To the end, authors first numerically found candidate solutions up to some ten decimal digits [2] and used specialized tools to prove (!) that there exists a real solution sufficiently close to given numerical solution. It turns out that there are several ways (and corresponding implementations) to do that, e.g. alphaCertified [3].

[1] http://arxiv.org/abs/1308.5164

[2] 10^-11 to be exact, according to the paper. It would be much easier to find a near-solution with larger errors (say, 10^-5). The physical model of such near-solution would be indistinguishable to the correct model.

[3] http://www.math.tamu.edu/~sottile/research/stories/alphaCert... (that actually does use GMP and MPFR for the obvious reason)

Interesting. I would have thought that one could solve such things exactly by representing each unique known irrational that arises (root 2, pi, etc) by its own rational multiplier, and then overloading the relevant equality checks. Of course, you'd need to anticipate/implement each irrational type that might arise (roots, the geometric transcendental pi, and so on.)

[Leaving the next sentence in, for comedy value. I typed it and then realised how ridiculous it is - apologies!]

Essentially it's just teaching the computer to do the algebra for you, isn't it?

How would you test whether two generic irrational numbers were equal? Obviously you can numerically approximate them and if you see any difference in the numerical approximation then they must be different - but if they seem the same up to e.g. 10 decimal places, what do you do next?

I would generally assume that sqrts of primes don't overlap, so you can always do exact comparisons of rational coefficients? It wouldn't be perfect, but it would do as well as a person with pen and paper and a hundred years.

[[ Approximate algorithm, in case I'm not being clear: you need sqrt(35), represent it as 1×sqrt(5) × 1×sqrt(7). You simplify each expression evaluated down to roughly what you'd write on paper in RAM, and then you do exact comparisons - is that the same coefficients of the same number of the same prime sqrts? For greater than/less than you cast them to a hundred-sig-fig float, and if those are still equal, keep going down the rabbit hole. Obviously the RAM requirements for complicated numbers would be large, but that's the same with rationals - this is just taking it to the extreme. I'd be very surprised if some function of sqrts was exactly the same as some other function of different sqrts. ]]

Sines of certain fractions of pi are equal to certain ratios of square roots though. And once you get to more complicated functions it's very hard to prove anything (e.g. I believe it's still not even proven that pi^e is irrational, never mind transcendental). Could we construct a countable field (for to be able to write down the numbers involved the field would have to be countable, i.e. only an infinitesimal fraction of the reals) that contained everything we need to do this kind of geometry in? Maybe. But I can just as easily believe you could construct a way of arranging objects that required a turing-complete computation to determine whether two objects met, in which case the halting problem makes this impossible.

what if irrational numbers are involved?

you couldn't argue with that

Plastic? I'm confused. The article states:

"Finally, they built a _wooden_ model to demonstrate their answer"

Links from one of the best riddle website : http://www.wuriddles.com/cigarettes.shtml

Cylinder Length | Max cylinders that can touch | Min cylinders that can touch

Infinite | 7 | 5

Actual | 9 | 7

L=D | 4 | 4

Those "max" values are speculation, while the "Min" values are for definite because there are constructions. So that table is of less value than you might think.

This is just one of many puzzles and mathematical curiosities popularized by Martin Gardner. His books and columns make delightful light reading for anyone with a curious mind.

Never seen this sort of thing before. Does this hold for an arbitrary radius? What if these were just lines in 3 space?

Arbitrary radius - just scale up and/or down as required.

If lines are considered cylinders, they would have a radius of zero and cannot touch without intersecting.

Intuitively, it seems that radius/length ratio is what's important here.

Yes, length to radius is important factor. Here's a discussion on 7 cylinders touching (using their end) ...


No, as this cylinders can be infinitely long if I'm not mistaken

Sure, there's no lower bound on the radius/length ratio.

But there is an upper bound (think huge radius, short length).

Yes - currently the best result for coins is 5 mutually touching. Neither beaten, nor proven to be maximal.

Something special about 7 cylinders or would this be equally hard/simple for 6 or 8 ?

They all touch each other - if you have a solution for 7, you already have a solution for 6 (just remove a cylinder).

It gets harder and harder (and then impossible) the more cylinders you have.

Now I wonder if there's a proof that it's impossible for 8...

There's an arrangement of 8 that's very close to touching, but has been proven to not actually touch: http://www.sciencedirect.com/science/article/pii/S0195669808...

Others believe they have found both an 8 and a 9: http://arxiv.org/pdf/1312.6207.pdf

though this may not be exactly the same problem. In particular, the 9 cylinder problem allows 3 radii to be selected and then the other 6 are calculated as a result (meaning the 9 are probably not all identical.) It appears the 7 result in the initial paper is 7 cylinders of equal radius.

The hippies of Marin County have known this since 1973.

Don't they just have to not be parallel?

No, your thinking 2D lines, but this is 3D space. It's easy to arrange things so they all miss each other and not have them be parallel. The goal is to have them touch, but not intersect.

In three dimensions, non-parallel lines do not necessarily intersect.

Yes, if the radius is infinite. But I think this is not the case.

Nah, not quite. You want every cylinder touching all 6 other cylinders and not going through them.

Proof the Internet is a series of tubes.

    they built a wooden model to demonstrate their answer — although Bozóki notes 
    that the model doesn’t verify the result because manufacturing errors 
    are much greater than any errors the computer could have made.
What's the point, when its not practically possible?

What's the point of what? Physical models have imperfections and distortions, so to absolutely know the answer you need the mathematics. Then having done the mathematics it's satisfying to build the physical model, to see it and to hold it.

Can you clarify your question? Try to avoid the word "it" because the referent might not be obvious.

The wooden model "works", eg they all touch. But it could be slightly less than perfect and you wouldn't be able to tell, mathematicians care about these things :)

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