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  • 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

  • indication that it might be optimal.

  • I mean, of course, you know our imagination

  • is limited. Maybe 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 am, myself, became interested in this

  • problem, more as a hobby then a some

  • kind of official research project I

  • start 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 what Gerver had done.

  • Because 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 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

  • was unable to do either of those things

  • so that was a bit discouraging. But then,

  • I had 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 to push it on through here and

  • now we get stuck because this is the

  • sofa that can only rotate to the right.

  • Now of course, when you have it in your

  • room and you were 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 ask myself

  • the question that is the natural variant

  • or generalization of the original

  • problem and actually turned out of 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?

  • [Ben Stiller as Derek Zoolander]: I'm not an ambi-turner.

  • it's a problem I've had since i was a baby. 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

  • defined shape it's perfectly good okay

  • so you push it 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

  • [Owen Wilson as Hansel]: There it is!

  • [Jon Voight as Larry Zoolander]: Holy Moley

  • [Will Ferrell as Mugatu]: It's beautiful!

  • [ Christine Taylor as Matilda Jeffries]: Derek you did it! That was amazing!

  • [Derek]: I know I turned left!

  • [Prof. Romik]: It's quite subtle, in fact it's subtle in many of the same