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

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

#31

Post by lotharbot »

ColinWright wrote: post #21

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 infin…

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.

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

#32

Post by yaketysax »

josephwegner wrote: post #25

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.…

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.

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

#33

Post by pavelrub »

josephwegner wrote: post #25

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.…

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".

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

#34

Post by Someone »

josephwegner wrote: post #25

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.…

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))

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

#36

Post by lifthrasiir »

tehwalrus wrote: post #22
delinka wrote: post #12

Earlier quoted context omitted.

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)

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

#37

Post by chubot »

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.

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

#38

Post by eridius »

ColinWright wrote: post #29
josephwegner wrote: post #25

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.…

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 regar…

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.

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

#39

Post by pisarzp »

acchow wrote: post #17
laxatives wrote: post #10

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

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

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

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

#40

Post by pisarzp »

acchow wrote: post #17
laxatives wrote: post #10

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

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

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