Mathematicians find way to put 7 cylinders in contact without using their ends

https://www.sciencenews.org/article/tale-touching-tubes

by ColinWright • 12 years ago

265 84 12 years ago

sciencenews.org

Zitrax• 12 years ago

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

SamReidHughes• 12 years ago

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

cousin_it• 12 years ago

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

acchow3• 12 years ago

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

ebol4• 12 years ago

Don't they just have to not be parallel?

Retric• 12 years ago

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.

anon4• 12 years ago

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

saalweachter• 12 years ago

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

lcnmrn• 12 years ago

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

josh-wrale• 12 years ago

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

kenferry• 12 years ago

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.

delinka• 12 years ago

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.

tehwalrus3• 12 years ago

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

[1] https://gmplib.org/

darklajid• 12 years ago

Plastic? I'm confused. The article states:

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

laxatives• 12 years ago

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

deckar01• 12 years ago

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

ColinWright8• 12 years ago

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

acchow3• 12 years ago

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

mythealias• 12 years ago

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

http://www.ocf.berkeley.edu/~wwu/riddles/cigarettes.shtml

pisarzp• 12 years ago

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

soneca• 12 years ago

Great design for a no-gravity space station.

Easy to go everywhere from everywhere.

aaronem3• 12 years ago

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.

gatehouse• 12 years ago

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

3rd32• 12 years ago

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

jzwinck• 12 years ago

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.

ColinWright8• 12 years ago

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.

lotharbot2• 12 years ago

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.

darkmighty2• 12 years ago

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.

josephwegner• 12 years ago

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.

ColinWright8• 12 years ago

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."

eridius• 12 years ago

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.

wesleyd• 12 years ago

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.

jafaku• 12 years ago

Brb, I'm gonna create CylinderCoin.

alister2• 12 years ago

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

yaketysax• 12 years ago

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.

pavelrub3• 12 years ago

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".

trevyn• 12 years ago

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.

Someone2• 12 years ago

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))

wolfgke• 12 years ago

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)

alister2• 12 years ago

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...

pbhjpbhj• 12 years ago

"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.

pjbrunet4• 12 years ago

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 ;-)

chrisdone• 12 years ago

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.

mathgenius• 12 years ago

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.

huhtenberg• 12 years ago

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

chubot2• 12 years ago

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.

pavelrub3• 12 years ago

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!

http://en.wikipedia.org/wiki/Klein_bottle

Never heard about the gomboc before, thanks!

anigbrowl• 12 years ago

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.

chubot2• 12 years ago

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

kang• 12 years ago

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

ColinWright8• 12 years ago

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.

ColinWright8• 12 years ago

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 ?

gary4gar• 12 years ago

    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?

almost• 12 years ago

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 :)

ColinWright8• 12 years ago

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.

thret• 12 years ago

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.

analog31• 12 years ago

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

darkmighty2• 12 years ago

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).

pjbrunet4• 12 years ago

Proof the Internet is a series of tubes.

DonGateley• 12 years ago

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.

owenversteeg• 12 years ago

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

pervycreeper• 12 years ago

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.

ColinWright8• 12 years ago

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.

abalone• 12 years ago

The hippies of Marin County have known this since 1973.

Mathematicians find way to put 7 cylinders in contact without using their ends

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