Perplexing Paperclips – Numberphile


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-chain—and 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 time—version 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 and—number 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 picture— my 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 numbers — or, perhaps at a more advanced level — calculating 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 band — the top one — as 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. Let’s try rubber bands in the position that we used to call 1+1, in other words: both rubber bands are linked with the paper. As far as the rubber bands are concerned, it’s just like 1+1. As you can see both rubber bands are linked to the paper; they have to stay on the paper. What’s different from the previous experiment is that one of the paper clips is at the top, but the other one is at the bottom. It used to be both at the top, and both at the top – mysteriously, they used to link. But this time, the paper clips will not be linked. Rubber bands, however – we have already understood the topology, because the paper goes through the rubber band – has to stay on the paper. So, what’s going to happen when I pull the ends of the paper? Huh? Any takers? Any guesses? BRADY: Like, maybe, each rubber band will have one paper clip? TADASHI: Okay, that’s interesting, so… My friend Brady’s saying, well, when I straighten out the paper, rubber bands will be hanging from the paper; that’s obligatory. Each rubber band will have a paper clip, this rubber band, paper clip. But paper clips will not link because they’re on the wrong sides of the paper. Let’s try this. Excellent. That’s what happens. You see? We have started computing. We understand something. There’s a lot of method to be smart in this. I guess, interesting, let’s try this. That’s inter— it looks like the previous one but this time, each paper clips is linked to the rubber band, but not between themselves.

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