Placeholder Image

字幕表 動画を再生する

  • TADASHI TOKIEDA: A strip of paper, a paper ribbon, and two ordinary paper clips.

  • We bend the paper strip in a Z or S shape, like so.

  • You know, going winding about like that.

  • And, with the help of the paper clips, we shall fix that shape

  • by slotting the paper clips here and here.

  • Now, what happens if I start pulling the ends of the paper?

  • Well, you can see that the paper clips come closer and closer together,

  • because they are being driven by the paper.

  • So, if I keep pulling, eventually they'll touch and jam,

  • but, if I keep pulling nonetheless...

  • they hop!

  • And, when I pick them up...

  • they're linked. That's quite interesting.

  • They're linked.

  • What we have just done is with two paper clips only, and this what we shall call 'version number zero'.

  • This is just the beginning. The trick goes back quite a long way;

  • according to Martin Gardner's Encyclopedia of Impromptu Magic,

  • this seems to have been discovered by somebody called Bowman in Washington State.

  • Today we'll do some variations on this.

  • So, we fix the shape of the bending paper like so.

  • But this time we added a rubber band. You see, the paper has two loops here and here,

  • and, seeing from your side, we're hanging the rubber band on the left loop.

  • This is version number 1. Version number 1, then, hanging over here.

  • What happens? This time what happens is this.

  • You have the rubber band hanging from the paper ribbon;

  • underneath clip-clip together from the rubber band.

  • We have this kind of half-chainand you will see in a moment why we call it 'half-chain'—

  • of rubber band-clip-clip together.

  • In science, you have to exercise your power of imagination and observation, especially.

  • You know, you can't just take something and say "Oh, this is not going to matter",

  • and then reject little differences of innocuous appearance.

  • Maybe you have to pay a lot of attention to such little differences.

  • To illustrate that point, let's do something that looks like version number 1,

  • which we have just tried, but isn't quite.

  • Number 1 that we have just tried looked like this

  • The rubber band was hanging from the left loop of those two loops.

  • Okay, now [clears throat]

  • The rubber band is now hanging from the right loop instead of left loop,

  • and that's what we shall call 'version number 2'.

  • You see, before it was hanging here, and now it's hanging here.

  • So, that's version number 2.

  • What's going to happen if I pull the ends of the paper?

  • Well, you know, somebody not observant might say "Well, what's the difference?

  • Here, here, well, probably doesn't matter."

  • But you remember that version number 1 ended up having the rubber band hanging from the paper ribbon,

  • with two paper clips hanging from the rubber band.

  • This timeversion number 2—whole thing falls off the paper ribbon,

  • so that's different. Nonetheless, when you pick it up,

  • you still get this clip-clip-rubber band, this half-chain together.

  • So, in the case of version number 1, we had this,

  • and, version number 2, this got detached.

  • Having tried version number 1 and version number 2,

  • it becomes tempting, indeed irresistible,

  • to try both at the same time, what is 1+2.

  • Well, it's very lucky that I have another rubber band here.

  • I think, and I'm sure you think too, 1+2 should look like this.

  • You see, because by itself, this rubber band is in the position of version number 1.

  • It's hanging from the left.

  • This one, by itself, is in the position of version number 2,

  • so 1+2 should be like this; it's doing 1 and 2 at the same time.

  • So, the question is what happens if I pull the ends of the paper.

  • What's going to happen to the two rubber bands this time, and the two paper clips?

  • Before we do that, however, let me debunk myself a little bit.

  • The fact that, in version number 1, the rubber band stayed on the paper

  • andnumber 2—the rubber band fell off the paper,

  • is, in fact, no mystery; we can understand this.

  • So, let's try to understand it--

  • by ignoring the paper clips.

  • You remember that version number 1 looked like this?

  • The rubber band was hanging from the paper on the left loop.

  • If you look carefully, you see that the paper strip goes through the rubber band

  • and then passes on the side.

  • In other words, the rubber band is linked with the paper strip.

  • So, if I pull the ends of the paper...

  • of course it stays linked, and it has no choice but to stay on the paper.

  • That is why, in version number 1, the rubber band stayed on the paper.

  • In contrast, version number 2 looked like this.

  • Over the two loops in the paper, it was hanging from the right loop.

  • If you look carefully, you see that the paper comes into the rubber band but goes back out.

  • So, the rubber band is, in fact, not linked to the paper.

  • When I pull the ends then, being unlinked, the band is free to fall.

  • That is why the rubber band can fall off the paper in version number 2.

  • So, the fact that the version number 1 stays on the paper, number 2 falls off the paper,

  • is no mystery as I say.

  • That's a purely topological thing that we can understand.

  • Let's go back then where we left off and try to do 1+2.

  • So, what's going to happen if we do 1 and 2 at the same time?

  • You should pause and try to guess.

  • BRADY: They're all gonna to be linked, and on the paper.

  • TADASHI: Good. But, a bit more precisely

  • BRADY: I picture one big long chain

  • TADASHI: Long chain. Rubber band-clip-clip-rubber band, in that kind of order?

  • Or rubber band-clip-rubber band-clip?

  • Or, maybe two rubber bands get linked? Well, I do that only for money.

  • BRADY: I don't know, I picturemy guess is rubber band-paper clip-paper clip-rubber band.

  • TADASHI: Excellent, OK, so that's quite nice. By the way, it's always important to guess anyway,

  • because, you know, guessing is really the way to learn and, in fact, advance in science,

  • both for students and researchers alike, because if you guess right, you are very, very proud you got it right.

  • And if you guess wrong, you are really shocked; maybe not really, but slightly shocked,

  • and that engages your thinking, and you can learn what happened,

  • and then it makes you a little smarter next time.

  • So, always guess before solving any problem.

  • OK, let's try doing this.

  • My friend Brady guessed rubber band-clip-clip-rubber band.

  • Let's do this. 1+2, what's going to happen?

  • Here we go.

  • Of course that's what happens, congratulations.

  • And it makes sense, in retrospect, because if we didn't have this rubber band

  • you see that configuration -- the remaining configuration -- is, of course, the number 1 configuration.

  • If we didn't have this rubber band, well, you get clip-clip-rubber band hanging -- that's number 2 --

  • but that would be because this rubber band is not present,

  • would be detached from the paper, so everything will fall down.

  • So, together, you get this long chain.

  • When we hear about computation, or calculation in general,

  • we always think naturally because that's how we learn these things in school, and about numbers

  • calculating with numbersor, perhaps at a more advanced levelcalculating with formulas.

  • And, you know, you do some algorithm, you do some recipe

  • and then the expected results come out.

  • But here, your brain, although it hasn't formalised anything,

  • you know, it hasn't written any numbers or formulas,

  • has effectively computed.

  • It has caught on to something- some really rich pattern in nature,

  • and has started understanding what was going to happen.

  • So, it's quite curious that your brain is good at computing,

  • even without numbers or formulas.

  • OK, what happens if I take two copies of number 1 on this paper ribbon,

  • and do the same experiment pulling the ends?

  • 1+1...

  • looks like this.

  • You see? This rubber band is in the position of now-familiar position number 1.

  • This rubber bandthe top oneas you can see by looking at it from the back,

  • is another copy of version number 1.

  • So, we have here two copies of version number 1 living together on this strip of paper.

  • And you can see that both of those rubber bands are linked to the paper

  • in the sense that the paper goes through one and through the other.

  • So, when I pull the ends, the rubber bands must stay on the paper, that much we know.

  • The only question is: how do they interact with the paper clips?

  • You care to guess?

  • BRADY: I think both rubber bands are gonna stay on the paper...

  • TADASHI: That's true.

  • BRADY: ...and the two paper clips will be linked to each of them on the ends.

  • TADASHI: How about paper clips between themselves?

  • BRADY: I think it'll be rubber band-paper clip-paper clip-rubber band.

  • TADASHI: Ah, that's good. Indeed, we are guessing that we- I can get that long chain of band-clip-clip-band.

  • Let's try this.

  • Indeed.

  • Of course.

  • Again, we thought nature should behave in a certain way, and nature obliged.

  • You get this kind of suspension bridge.

  • Rubber bands on the paper, that we had already understood topologically,

  • but what is new is that you get, again, this long chain of band-clip-clip-band

  • hanging like this; paper clips connecting the rubber bands in the middle in this hanging configuration.

  • That's 1+1. So...

  • 1+2 was like this because number 1 is linked to the paper; number 2 isn't.

  • And 1+1 was like this because they are both copies of number 1 and they have to be on the paper.

  • Each time, then, we are getting a long chain- a full chain of band-clip-clip-band,

  • and the only question is how that long chain is positioned with respect to the paper.

  • Well, it remains for us to try 2+2...

  • Version number 2, I'd like to recall, is the one where by itself fell off the paper.

  • 2+2 is like this and that's because- if you look at, for example, just at this rubber band

  • that's in the now-familiar position of number 2;

  • paper goes in but comes back out.

  • And over here, at the top, you will see another copy of version number 2.

  • OK, so, what we have to do is to pull the ends and see what happens.

  • Well, this time, neither of the band, by itself, is linked to the paper.

  • So, what's going to happen?

  • When I was exploring this, and trying various possibilities and some,

  • at this stage, when I came to this stage for the first time,0.

  • naturally, I guessed that I would get a long chain of band-clip-clip-band,

  • which we do.

  • And because neither band is linked to the paper, I thought the entire thing will just fall down to the ground,

  • leaving the paper alone.

  • So, let's try this.

  • This time, however, something strange has happened.

  • The entire ensemble of the rubber bands and paper clips didn't fall down to the ground;

  • instead, they stayed on the paper.

  • You see? There is a rubber band that goes around the paper like this in a transverse manner.

  • And then there is - in the middle - a rubber band, which is kind of acting as a lock, so to speak.

  • Please ignore the paper clips there; not that interesting at this stage.

  • So, there's a middle "lock", and then the band that goes around - transversely - around the paper.

  • You can see that if we didn't have this rubber band, the other one could be pulled off the paper.

  • On the other hand, if we didn't have this rubber band,

  • then this middle lock, which is hanging, would fall down to the ground.

  • So, neither band is linked to the paper, that's true,

  • but each one is preventing the other from leaving the paper.

  • So that, if I now closed the ends of this paper,

  • and think of the paper as the third loop, we have here a system of three loops.

  • Any two of these are mutually unlinked, but three together...

  • are linked and stuck together.

  • And that kind of configuration has a name: it is called a Borromean link.

  • Borromean link,

  • after the Italian Renaissance family called Borromeo

  • no, no, it's not the family who used to poison one another; that's Borgia, that's another family

  • but Borromeo.

  • And it appears in all sorts of contexts.

  • Very, very important central object, for example, in 3-dimensional topology.

  • Here is an example of a Borromean link.

  • Each of those three pieces of wood is a loop

  • rather, sort of, distorted loop, but it is a loop nonetheless

  • and they're clearly stuck together; you cannot pull them apart.

  • But, you see, it's curious to note that, if you make in your mind's eye

  • any one of these three disappear, by saying "kazam", for example.

  • Let's make the one that I'm holding disappear.

  • Then the other two can be pulled apart, so they are not linked.

  • And it's the same for any of these three.

  • So, if I make this one disappear, the other two can be pulled apart, and

  • any one of these disappearing will cause the other two to fall apart.

  • So, that is a Borromean configuration.

  • This is carved out of a single piece of wood, and when I first bought it in Africa,

  • I was very impressed to hear this.

  • But then, you caught me, that that's the only way to carve something like this, of course.

  • Good, we have seen quite a few things,

  • and if you wanted to share this with friends and family, I suggest that you end on the following,

  • rather mischievous, note.

  • You say "OK, OK, folks, you understand everything, right?",

  • and by this time, they're feeling cocky and confident.

  • "Yeah, yeah, we understand everything", and you say "OK, let's revise the first thing that we started with".

  • What happens if I bend the paper like this in a Z or S shape,

  • and, with the help of paper clips, fix them in place, and then pull the ends of the paper?

  • They say "Oh, we have seen all that", and "they get linked together, right?", and so on.

  • It's really strange because, not more than ten minutes ago,

  • they had never seen paper clips come together. Now they assume that, every single time, they come together.

  • That's quite curious.

  • So, you say "OK, OK, but let's try this."

  • So, paper clips on paper, and when I pull - ah, that's strange; they didn't get linked together.

  • What happened?

  • 'Course, some people - maybe not you - were distracted by my patter,

  • and haven't been watching.

  • What we did was, indeed, put the paper clip at the top of this Z shape.

  • But the other one - instead of putting it at the top, I put it at the bottom on the other side.

  • So, paper clip here...

  • and here.

  • So, when I pull, they're completely separated; there's no incentive for these to link together.

  • So, you have to keep watching.

  • But not all is lost. This is just a piece of mischief, but you can start doing science on top of this.

  • Let's try the paper clips in the wrong positions, but with addition of the rubber bands.