Subtitles section Play video Print subtitles 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 - 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 simple situations in capturing that would be when you have a turn, a right turn, in the 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. [Brady]: Largest area, not longer [?] [Prof. Romik]: Not longest, not heaviest, just largest area. [Brady]: OK. [Prof. Romik]: not most comfortable So here's an example of one of the most simple sofas you can imagine so it has a semi circular 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. [Brady]: and what's the area of that one? Like is that a good area? [Prof. Romik]: First of all we have to say that we choose units where the width of the corridor is one unit let's say one metre or something like that then the semicircle have radius one so I'm sure all your viewers know that the area would be PI over 2 because that's the area of a semi circle 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 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. I'm so you see you can push it and again it meets the wall and now we 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 semi-circular one and you should imagine cutting up the 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 semi circular in shape then everything will work the way it should and so it can move around the corner and he also optimized that particular parameters associated with how far apart you want to push the two quarter circles and so on. And then you work out the area of the overall area of the sofa and it comes out to two pi over 2 plus 2 over pi. So 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 20 something 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 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 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 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. [Brady]: Cool [Prof. Romik]: And, well, let's see it in action. [Brady]: Yeah! [Prof. Romik]: Okay so we put it here we push it and you see, I mean it looks roughly the same as what happens with Hamersley's sofa, except the 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 is 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, a mathematician from Rutgers University. The area of his sofa is 2.2195 roughly so about half a percent bigger than Hammersley 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 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. [Brady] Did he conjecture based on anything of rigor or was it just he came up with so he's affected he's fond of his desire. [Prof. Romik] Um, well it could be that he's fond of his design I have no doubt. Um, nobody has some real 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 sofas 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 the function then to find a 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 be optimal, so, and it's the only sofa that has been found that satisfied to this necessary condition so that's pretty good