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Boring Math Stuff

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09:40 / 21.03.02

I'm in two minds about whether to start this thread, especially given the presence of journos here. I do want to make it clear that this is all a bit tentative and really "cutting edge". So it might bollocks-up come tomorrow. No guarantees, no certainties. Have I made this clear? Have I been dull enough yet?

Hopefully, the only people left are the anti-thrill seekers. Well, here we go.

There are some maths problems out there that are big. Quite big. Fermat's Last Theorem was one. That was done, after a very long time. Another less famous problem is Poincare's conjecture. This is a really key three dimensional geometry problem. Its a bit complex to explain, but here is a link. Its about our intuition for three dimensional things. Kinda. And its very hard. And its worth a million dollars. And someone in my department has just solved it. Probably. It'll take a few months (at least) to be sure, but the signs are good. Fingers crossed.

Less searchable M0rd4nt

09:48 / 21.03.02

I knew you couldn't resist it.

09:48 / 21.03.02

I tried to make it as dull as possible, at least.

Less searchable M0rd4nt

09:48 / 21.03.02

You know, I hope this guy did solve it. He sounds nice. The way you describe him, I'd get a real kick out of his being the one to solve Poincare's Conjecture (not to mention the winning a million quid part).

09:48 / 21.03.02

quote:If we stretch a rubber band around the surface of an apple, then we can shrink it down to a point by moving it slowly, without tearing it and without allowing it to leave the surface. On the other hand, if we imagine that the same rubber band has somehow been stretched in the appropriate direction around a doughnut, then there is no way of shrinking it to a point without breaking either the rubber band or the doughnut.

hah? I don't geddit - shrink the rubber band down to a point? how do you do that?

'splain! 'splain!

damn mathematicians and metaphors...

m.

09:48 / 21.03.02

Tricky.

The point is that you are allowed to make the rubber band smaller, but you aren't allowed to break it. (Call these the rules of some game.)

If you do this on an apple, then as you slowly shrink it, it becomes ever smaller until it becomes a point.

If you do it on a doughnut, then sometimes the same will happen. But if you imagine the rubber band round the doughnut in certain positions (round the short circular bit in cross section or round the long cirular bit in circumference) then you can't shrink the rubber band to a point without breaking the doughnut or making the rubber band leave the doughnut.

BTW - for our purposes, a doughnut should be hollow and properly called a torus. Doesn't give you much filling, I know...

09:48 / 21.03.02

I've re-read the quote by matsya and I realise I haven't said any more than the quote. In fact, the quote gives a pretty good description of what you should imagine.

Less good, but perhaps easier is to try to imagine a computer animation of a rubber band on this apple becoming a point. Think of how you would animate this (including all the intermediate steps) with no sudden "jerks". You can always do it. On a doughnut, you can't.

12:02 / 21.03.02

I'm interested, but need more explanation. Can you first clarify the anture of this shrinking business? At first i had thought that we were referring to some strange intangible apple (or sphere), and that the rubber band would shrink toward its centre. But that ain't it, right? you are sliding the band along the surface in a way that, if we assume its original position was the equator, it will shrink to a point at either pole, assuming you're moving it at equal speed along each longitude. (Sorry, I've slipped into using a planet as a metaphor, nut hey, it's all abstract modelling, right?). But I don't get why the torus or band would have to break. I can see why it can't be reduced to a point in smooth movements (computer animation simile v helpful there, Lurid), but I picture it having to jump across the hole. Ping! That's it, isn't it - it either has to leave the surface or one of the objects has to break. So the hole is the problem - a rubber band over a cube, or the Venus de Milo, could still be reduced to a point.
Now that I (think) i understand the problem, I'll go look at the solution.

12:07 / 21.03.02

oh god, hang on - are we imagining the band around the doughnut like a garter on a leg, or around the circumference of the whole thing, including the hole in the centre, like, for example, the seam of a rubber ring (this is what I was imagining in my previous post)? or does the problem encompass any possible situation?

12:21 / 21.03.02

Lentil: Sounds to me like you understand it perfectly. That was very quick. I think it took me a few weeks to understand it.

Your comments about the cube and venus de milo are excellent - a born topologist!

Yes, the apple should really be hollow and is properly called a sphere or 2-sphere if you want to be pedantic. The band should slide along the outside. The equator to poles picture is bang on the money, though speed doesnt really matter.

As for the doughnut. This has two different types of holes that can't be shrunk. The "garter" and the "circumference". If you look at the doughnut in plan view, you see two types of circumference. A sort of inside and outside. The rubber band can move smoothly between them.

The point is that for the apple, any rubber band in any position can be smoothly shrunk. For the doughnut, there are some that can't. This distinguishes them in some sense. You can be more precise about the fact that the doughnut has "essentially" two configurations. I've simplified a bit, of course.

For those who like jargon: An object like the hollow apple, where any rubber band in any position can be smoothly shrunk to a point is called "simply connected". It tells you something about the absence of holes.

13:58 / 21.03.02

I'm holding an apple in my hand.

I'm putting a rubber bandaround it's equator.

Hmm... The rubber band isn't getting any smaller. But if I move the rubber band up or down then it shrinks (if by shrinking you mean constricting.)

Now I take a doughnut. I put the ruberband around the circumfrence of the doughnet. Then I drag it off. It constricts the moment it comes off the doughnut.

So, that's why I'm lost. I have no idea what you're talking about.

14:02 / 21.03.02

I was going to post right back but was interrupted on some buearaucratic nonsense. so was Poincare saying that any three dimensional solid without any holes exhibits the same characteristics in relation to the hypothetical rubber band as our apple/sphere/planet? And if so, has your colleague proved or disproved his conjecture? Or can't you say?

14:03 / 21.03.02

Yeah, its confusing. Partly because its not a real rubber band we are talking about. For "shrinking" to take place, the rubber band has to get smaller! And, for the rules of the game, it has to always stay on the surface of the apple or doughnut. No part can come off.

Try thinking of that 3D computer animation that Lentil describes, shrinking from the equator to a pole.

14:06 / 21.03.02

Should really leave this to Lurid Archive, but Lionheart, if I'm understanding this properly...

Firstly, you kind of need to imagine that the rubber band is magic, and can contract infinitely. Secondly, the problem is not just whether or not the rubber band can contract to a point, but whether it can do it without breaking its contact with the surface of the object. Does that make any sense?

14:08 / 21.03.02

Arg! You beat me to it! What about my other question above, just after Lionheart's post?

14:36 / 21.03.02

quote:Originally posted by a Man Called Lentil:
so was Poincare saying that any three dimensional solid without any holes exhibits the same characteristics in relation to the hypothetical rubber band as our apple/sphere/planet? And if so, has your colleague proved or disproved his conjecture? Or can't you say?

Thats not quite right....

The simple way to say it is that the only 3D object with no "holes" in it is a solid rubber ball.

The holes here can be of any dimension. So the hollow apple has a two dimensional hole, but no one dimensional holes and the doughnut has one dimensional holes.

There is another, more usual, explanation that relates more closely to the rubber band stuff but its a bit more tricky to explain. I'll try it in the next post.

PS My colleague still believes he has proved the Poincare conjecture. So he should win the million dollars and be the subject of a horizon doc if all goes well.

14:48 / 21.03.02

A sphere (hollow apple) is two dimensional because as we all know, if you are small and standing on one it looks like a flat piece of paper. (thats why people thought the earth was flat).
One way of imagining "building" a sphere is to take a sheet of paper and glue all the outside edge together without crumpling. Either that or take two sheets of paper and glue them along an edge. Notice that the sphere is two dimensional, but you need three dimensions to realise it.

Now you can do the same to build a 3-sphere. Take a solid cube and glue all the outside together to one point without crumpling. This is almost impossible to imagine (I can't properly). It lives in some 4 dimensional space in the same way that the hollow apple lives in 3 dimensional space.

Poincare's conjecture: Any "proper" 3D object that is simply connected (you can shrink those rubber bands) is the 3-sphere.

NB. The solid rubber ball I talked about in the above post isnt a proper 3D object in this sense because it has a 2D "edge". ie, the outside.

Just as a square (finite) sheet of paper isnt a "proper" 2D object because it has a 1D edge.

These two statements, in this and the previous post, turn out to be the same.

16:11 / 21.03.02

nice one. I didn't get the thing about dimensions before, I'm on the same page as you now.

06:54 / 22.03.02

im still in remedial over here, so let me try to get it down myself.

quite simply, you can turn the band on the doughnut to a point because as you slide it up, when it constricts maximally, it is in the hold, and no longer touching the doughnut's surface. is this right?

maybe to make it even easier to understand, we could say the rubber band WANTS to be as small as possible, but the goddamn material it's wrapped around keeps it from constricting. so it moves around -- "northward" contracting as it goes.

so perhaps what youre saying in a SENSE is that the rubber band is a test of a topological objects dimensionality (to sort of make up or misue a word); in the case of the sqhere it has a 0-dimensional possibility, as seen in the rubber band; but not so in the doughnut, whose rubber band can only become topologically one dimensional. is this right? so there is a topological qualitative difference between these two objects,

and as any ant knows, a teacup is topologically a doughnut.

08:07 / 22.03.02

Actually this confirms my gut feeling. A lot of people out there are interested in math and have a remarkably good feel for it - they end up hating it because all the life is sucked out of it at school.

Mystery Gypt: I think you've got a good grip on what is going on. Yes, for this game the band does want to be small and is prevented from doing this on a doughnut.

This rubber band lark does measure some sort of dimensionality - it says that the doughnut has one dimensional holes and the apple doesnt have any.

You can actually play the same game in all dimensions. In dimension two you place a hollow rubber ball on an object and try to shrink it....of course, its pretty hard to imagine. Unsurprisingly the hollow apple has a two dimensional hole but less clear is that the hollow doughnut doesn't.

08:11 / 22.03.02

BTW - anyone who has followed and understood the rubber band stuff should be very pleased with themselves. You've understood the key ideas involved in a masters course on "Algebraic topology".

anyone who doesn't get it, don't worry. its hard!

08:52 / 22.03.02

ok so what would a chart of topoligcal shapes to dimensional holes look like?

3D
sphere - no 1 dimenionsal holes
dougnut - 1 dimensional holes
?? -- ?? n-dimensional holes

4D
[i assume in your last example, you are putting the 3d sphere on 4-dimensional objects, asking to find 2 dimensional holes? is this right? and then you need the 4 dimensional equivalent of a doughnut? uh-oh. ]

this is great fun. i've always liked topology, but i've never studied it at all. but i do know you can't comb a fuzzy sphere smooth.

10:02 / 22.03.02

The two sites at the top of the google search are quite interesting, if more jargon-filled than Lurid Archive's explanation.
Thanks for that btw, I now understand enough to get something out of these:

Official problem description page

More fun, still jargonated history
(work in progress but good)

Can you say where your colleague works?
I can't see a danger in releasing this info as someone tends to claim the Poincare Conjecture fairly often, its not really news until independently confirmed.
- l

11:17 / 22.03.02

Mystery Gypt: The sphere and doughnut etc are usually called surfaces and there is a classification of what they can look like. A reasonable description is given here. Don't worry too much about the non orientable ones. They are usually thought of as 2-dimensional even though they live in 3D space.

luminocity: I am slightly worried about giving away too much info since my colleague is very media shy, but I think its sort of out at the moment. We are at southampton Uni and there is even a preprint on the server.

However, as of this morning there is a problem with the proof. (In fact this is about the fifth problem found so far, though all the previous ones were, in some sense, addressed.) The status of the result seems to change hourly. Its not dead in the water yet and I remain relatively optimistic about it.

00:38 / 23.03.02

Thanks.
Keep us posted, ok? best of luck to everyone involved. It's so cool when (beautiful) results like this get proved, I hope your mate gets it.
- l

01:13 / 23.03.02

...anyone who doesn't get it, don't worry. its hard! – Lurid Archive

Thanks

although I suspect it's much easier to see in the mind than to explain in words...A doughnut (I'm thinking the regular type and not Jelly or Bavarian Creme) is more visually akin to a slice or segment of a hollow sphere, which may be why the rubber band reacts differently.

01:26 / 25.03.02

okay i think i got the shrinking rubber band bit down now. the apple's simply connected, the torus-doughnut isn't. fine.

now, what's your friend think they've found? a way to prove that a torus-doughnut IS simply connected?

head... hurts... must... read... superhero... comics...

m.

06:36 / 25.03.02

This quote from Lurid Archive on the first page explains this pretty well, i think - helped me figure it out ,anyway:

quote:One way of imagining "building" a sphere is to take a sheet of paper and glue all the outside edge together without crumpling. Either that or take two sheets of paper and glue them along an edge. Notice that the sphere is two dimensional, but you need three dimensions to realise it.
Now you can do the same to build a 3-sphere. Take a solid cube and glue all the outside together to one point without crumpling. This is almost impossible to imagine (I can't properly). It lives in some 4 dimensional space in the same way that the hollow apple lives in 3 dimensional space.

Poincare's conjecture: Any "proper" 3D object that is simply connected (you can shrink those rubber bands) is the 3-sphere.

So if you took any 3d object which doesn't have any holes, such as the apple, or other examples given earlier like the Venus de Milo or a teacup (it would have to be a beaker I guess, as the handle creates a hole), and make this impossible-to-imagine 4d object with it by glueing every part of its surface to a point, in every case the object would be the same, the "3-sphere" mentioned above. Or so Poincare said, and Lurid's colleague has (fingers crossed) proved it.

11:09 / 25.03.02

Thats right Lentil.

An equivalent formulation is this:

Take a hollow apple and fill it in with three dimensions so that the resulting object has no "holes". You "obviously" get a solid apple. Poincare's conjecture is that this really is the only way to fill in that hollow apple.

I should mention that there are lots of strange, counter intuitive geometrical objects (eg Klein bottles) that one can construct mathematically, so what seems at first obvious becomes less so with some experience. Anyway, everyone knows that the "obvious" things are the hardest to prove.

As of today, the proof has undergone several minor alterations and currently stands as correct.

01:59 / 28.03.02

the 3-sphere - that's the same thing as a hypersphere, yes?

i read Hyperspace by michio kaku recently and he talked about a hypercube in sort of similar ways. i think.

m.

14:08 / 28.03.02

Yes, the 3-sphere is the same as the hypershere, but you have to be careful. Getting the hypercube from the cube is easier than getting the hypersphere from the shere. This is because it has to curve in on itself. Ie. if I go from a circle and try to go up a dimension I might end up with a cylinder rather than a sphere.

One more point. Its tempting to think of the hyerpsphere as a 4D object, much like the hypercube. This is correct since the hypersphere lives in a 4D world. However, if a 4D woman were to stand on a large hypersphere, she would think she was standing on a 3D space. Just as standing on the sphere looks like standing on the plane.

Anyway, a quick search on Google gives a good few sites explaining the hypersphere in detail.

04:04 / 02.04.02

so how is the proof standing?

02:11 / 05.04.02

still good.

15:22 / 16.04.02

Looks like your man made NPR this morning. (Note: that's a RealAudio file. The show's page is here.)

18:42 / 16.04.02

This is probably the right time to point out that after going to the british math conference, I saw a lot of scepticism about the result. some of it is clearly well founded and martin dunwoody now acknowledges a problem with the proof. he is on holiday at the moment, so if there is a fix it won't appear for at least a week or two.

I'm starting to feel a bit pessimistic about this now, but I don't underestimate Martin. He is an extremely sharp guy, if a little idiosynchratic.

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