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

ways the Gerver sofa is subtle so 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 and

in a few of them are circular arcs so

that's not very complicated but the

other ones are really pretty--pretty

complicated to describe the curve you

can write down formulas for them and

everything except there's some numerical

constants that are involved that you

can't write formulas for because they're

sort of we are obtained numerically by

solving certain equations now with the

new sofa that I discovered and I did

that by applying the same idea that

Gerver had developed and that I sort of

developed slightly further was led to in

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 something new.

[Brady]: it seems to have like, pointy ends, the ends seem pointed

[Prof Romik]: the ends are pointed yes they made a certain

angle and that angle is an interesting

numerical constant that also shows up in the analysis.

[Brady]: what's the angle?

[Prof Romik]: something like sixteen point six degrees and--but

more interestingly it has the 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 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 squared equals two 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 arc cosine of

thing or it's pi over 18 or something

like that so it's not easily expressible

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.

[Brady]: as Gerver's sofa is thought possibly to be the

optimal solution is your optimal

solution here for the ambi-turner-- [Prof Romik]: yes

[Brady]: proven to be optimal or you don't know

[Prof Romik]: no I don't know it so the state of

affairs is precisely the same as with

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 was optimal

that wouldn't surprise me in the least

and if they show it wasn't optimal and

that would surprise me a little bit okay

that's that's a good question because

there's a bit of a story there so what

happened was that I was playing with

this problem for several months actually

as a little bit of the hobby that

something served not to do with with my

normal research and had more to do with

my hobby of 3D printing.