The Moving Sofa Problem - Numberphile

Key Takeaways

So, we're going to talk about a problem in geometry and it's called the moving sofa problem. So, the problem is inspired by the real life problem of moving furniture around. It's called it's named after sofas, but it can be anything really. You have a piece of furniture you're carrying down a corridor in your house or down some whatever place and you need to navigate some obstacles. So, one of the simplest situations in capturing that would be when you have a turn, a right turn in a corridor. You need to move the sofa around. We're modeling this in two dimensions. So let's say the sofa is so heavy you can't even lift it up. You can only push it around on the floor. Obviously some sofas will fit around the corner, some will not. And people started asking themselves at some point, what is the largest sofa you can move around the corner? So that's the question. What is the sofa of largest area? Largest area, not longest or not longest, not heaviest, just largest area. Okay, not most comfortable. So here's an example of a one of the most simple sofas you can imagine. So it has a semic-ircular shape and we push it down the corridor. So let's see what happens. We push it until it meets the opposite wall. And now we rotate it. And of course because it's a semicircle, it can rotate just perfectly. And now it's in the other corridor. So you can push it forward. And what's the area of that one? Like is that a good area? First of all, we have to say that we choose units where the width of the corridor is one unit. Let's say 1 meter or something like that. Then the semicircle will have radius one. So, I'm sure all your viewers know that the area would be pi /2 because that's the area of a semicircle with radius 1. Now, whether that's good or not, that's that's up to you. It's not the best that you can do for sure, but it is what it is. So, the next one that I have here looks like this. So, it's still a fairly simple geometric shape and it was proposed by a British mathematician named John Hammersley in 1968. By the way, I should mention that the problem was first asked in 1966 by a mathematician named Leo Moser. Let's first of all check that it works and then I'll explain to you why it works. Um, so you you see you can push it and again it meets the wall and now you start rotating it but while you're rotating it you're also pushing it. So you're doing like this and it works perfectly. Now the idea behind this Hammersley sofa is you go back to the previous one which is the se the semic-ircular one and you should imagine cutting up this semicircle into two pieces which are both quarter circles and then pulling them apart and then there's a gap between them and you fill up this gap. Now, in order to make it work so that you can move it around the corner, you have to carve out a hole because that's what you need to do the rotation part. And Hammersley noticed, and this is a very simple geometric observation, is that if the hole is semic-ircular in shape, then everything will work the way it should. And so can move around the corner. And he also optimized the particular parameters associated with how far apart you want to push that two quarter circles and so on. And then you work out the area of this the the overall area of the sofa and it comes out to to pi over two plus two over pi. So a slightly more exotic number. Definitely an improvement, right? Well, that wasn't the end of the story. As it turns out, Hammersley wasn't sure if his sofa was optimal or not. He thought it might be. People shortly afterwards noticed that it's not. And only 20some years later, somebody came up with something that is better. It's not really dramatically better because the area is only slightly bigger, but it's dramatically more clever, I would say. So, this is a construction that was discovered later in ' 92 and it is uh it looks very similar to the sofa that Hammersley proposed, but it's not identical. So, it's subtly different from it. Well, here you see this curve is a semicircle, right? Here we're doing something a bit more sophisticated. So, you see we've polished off a little bit of the the sharp edge here. And also this curve is no longer a semicircle. It's something mathematically more complicated to describe. And this this curve on the outside here is no longer a quarter circle. In fact, it's it's a curve that is made up by gluing together several different mathematical curves. So this shape is quite elaborate to describe. The boundary of it is made up of 18 different curves that are glued together in a very precise way. Cool. And well, let's see it in action. Yeah. Okay. So, we put it here. We push it and you see I mean it looks roughly the same as what happens with Hammersley so far except a small difference here is that you have a gap now because we've carved off this piece. So there's a little bit of wiggle room here at the beginning. You can push it in several different ways. There's no unique path to push it. But anyway, if you push it, you see that it works just the same as before. By the way, this was found by a guy named Gerver. Joseph Gerver is a mathematician from Rgur's University. The area of his sofa is 2.2195 2195 roughly. So about half a percent bigger than Hammersley's sofa. A very small improvement. But like I said, mathematically it's a lot more interesting because the way he derived it was sort of by thinking more carefully about what it would mean for a sofa to have the largest area. It's not just an arbitrary construction. It's something that that was carefully thought out and you know leads to some very interesting equations that he solved. and he conjectured that that this sofa is the optimal one, the one that has the largest area and that is still not proved or disproved. So that's that's the open problem here. Did he conjecture based on anything of of rigor or was it just he came up with it so he's affect he's fond of his his design? Um well it could be that he's fond of his design. I have no doubt. Uh no but he had some re some some pretty good reasons to conjecture that it's optimal because like I said the way it was derived is by thinking what would it mean for sofa to be optimal in particular it would have to be locally optimal meaning you can't make a small perturbation to the shape like near some specific set of points that would increase the area. So I mean that's a typical approach in calculus when you're trying to maximize a function. Then to find the max the global maximum you often start by looking for the local maximum. Right? So that's kind of the reasoning that guided him. You could say that the sofa satisfies a condition that is a necessary condition to to be optimal. So and it's the only sofa that has been found that satisfies this necessary condition. So that's pretty good indication that it might be optimal. I mean of course you know our imagination is limited. Maybe we we just haven't been clever enough and haven't been able to find something that works better. But that's the best we can do. So recently I um myself became interested in this problem more as a hobby than as some kind of official research project. I started tinkering with it and trying to wrap my head around some of the math that goes into it which is surprisingly tricky. But interestingly I was able to find some new advances in sofa technology you could say. I did several things. The first thing I tried to do is to get a good understanding of what Gerver had done because it it really wasn't obvious. I mean, I was reading his paper and it's kind of pretty technical and dense. What can I do next? I mean, how can I improve on what he had done? And of course, two two obvious choices would be to try to find a better sofa than he did or to try to prove that you cannot find a better sofa. And sadly, I wasn't able to do either of those things. So that was a bit discouraging. But um then I had a an interesting idea to do something that is essentially a variation of what he had done. If we go back to this thing with the the house with the two corridors, right? Now imagine that your house has a slightly more complicated structure to it. What if it looks like this? So you have a corridor and then a turn and then another corridor and then another turn and another corridor. Let's see what happens when we try to put I mean even the simplest one of these sofas through this corridor. Right? So, we push it on through here. We rotate it. We push it on through here. And now we get stuck because this is a sofa that can only rotate to the right. Now, of course, when you have it in your room and you're sitting on it, that's not really it doesn't bother you. But for the purpose of transporting it, that can be a nuisance, right? So then I asked myself the question that is sort of a natural variant or generalization of of the original problem and actually it turned out that this was a version of the problem that had been thought about by other people as well and I refer to it as the ambidextrous moving sofa problem. So this is to consider all sofa shapes that can move around this corridor meaning so they can turn in both directions and out of that class of sofas to find the one that has the largest area. So, you're looking for the optimal ambi turner? Have you seen the film Zoolander? I'm not an ambi turner. It's a problem I had since I was a baby. I can't turn left. Well, then I ended up finding actually a new shape that that satisfies this condition of being able to turn in both directions. It no longer looks very much like a realistic sofa, but mathematically, of course, it's a well- definfined shape. It's perfectly good. Okay. So you push it, you you rotate it while pushing it and it works. And of course it's going to work equally well in the other direction because it's symmetric so it doesn't distinguish left from right. There it is. Holy moly. It's beautiful. Derek, you did it. That was amazing. I know. I turned left. It's quite subtle. In fact, it's subtle in many of the same ways that Gerver's sofa is subtle. So if if you remember I told you that to describe Gerver's sofa you need 18 different curves. I mean there's three of them are that are just straight line segments but the other 15 are just are curved. I mean a few of them are are circular arcs. So that's not very complicated but the other ones are really pretty pretty complicated to describe curves. you can write down formulas for them and everything except there are some numerical constants that are involved that you can't write formulas for because they are sort of they are obtained numerically by solving certain equations. Now with the new software that I discovered and I did that by applying the same ideas that Gerver had developed and that I sort of developed slightly further was led to a certain system of equations that I had to solve and I solved it and that's where the shape comes from and again it turns out the shape is made of 18 different curves that you need to glue together in a very precise way. So yes, it is definitely quite elaborate. It's not like it's not a circle. It's not a square. It's it's something new. It seems to have like pointy ends. The ends seem pointed. The ends are pointed. Yes. They meet at a certain angle and that angle is an interesting numerical constant that also shows up in the analysis. What's the angle? Um something like 16.6° and but more interestingly it has a precise formula that I can write down for you. And this is another big surprise that I had when I found this with like I said with Gerver's sofa it you can describe it. I mean there's a full description of what Gerver's sofa is, but in math we we like to distinguish between things that can be written in closed form and things that can't be written in closed form. So a number like square root of two is a number that you can write in closed form, right? Of course, that's just shorthand for saying it solves the equation x^2= 2. But there are numbers that come from solving like a system of maybe two or three equations. And there isn't a simple way to say this number is the the arc cosine of something or it's pi over 18 or something like that. So it's not easily expressable in terms of known constants. And that's the feature that Gerver's sofa has is that to describe it properly you need to put in certain numerical constants that that cannot be written in closed form. Whereas when I found my new shape I discovered that it can be written in closed form. In fact, all the equations that describe it are algebraic equations. Not not something I was expecting at all and makes everything in some sense nicer. As Ger so far is thought possibly to be the optimal solution. Is your optimal solution here for the ambi turner? Yeah. Proven to be optimal or you don't know? No, I don't know. And so the state of affairs is precisely the same as with the the original problem, namely that nothing is proved about what shape is optimal. But I derive the shape that is a good candidate to be the optimal. I mean I'm not going on record as you saying this is a conjecture of mine because I don't feel confident enough to make such a conjecture but certainly it would be a very plausible candidate and if somebody were to come and show that it's optimal that wouldn't surprise me in the least and if they show it wasn't optimal then that would surprise me a little bit. Okay that's that's a good question because there's a bit of a story there. Um, so what happened was that I was playing with this problem for several months actually as a little bit of a hobby that something sort of not to do with with my normal research and had more to do with my hobby of 3D printing.