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.