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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

#81

Post by ColinWright »

DonGateley wrote: post #73

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.

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

#83

Post by mathgenius »

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.…

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.

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

#84

Post by chubot »

chubot wrote: post #37

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.

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

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

#85

Post by pbhjpbhj »

alister wrote: post #54
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.…

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.

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

#86

Post by Someone »

anigbrowl wrote: post #78
pavelrub wrote: post #43

Earlier quoted context omitted.

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!

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

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

#87

Post by lmm »

tehwalrus wrote: post #66
lmm wrote: post #59

Earlier quoted context omitted.

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…

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.
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