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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.

• 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