- Thank you to Brilliant for sponsoring this episode.
Okay, picture this, you've just moved
into your dream home complete
with an intriguing 90 degree angled hallway.
Your heart is set on creating the perfect cozy nook
for Netflix binges with friends, but there's a twist.
Literally, you need a massive sofa
that can navigate this perplexing hallway.
What is the largest sofa that can make the journey?
What does it look like?
On the surface,
this appears to be a straightforward geometry problem
but it's still unsolved to this day
despite mathematicians working on it for decades.
Why is this problem so hard to solve?
And what's the biggest sofa we've come up with so far?
- K Birds.
K birds.
- Shut up, shut up, shut off.
- Here on Up And Atom, we ask the important questions.
The origin of the moving sofa problem can be traced back
to Austrian Canadian mathematician
Leo Moser's 1966 publication.
The whispers
of its existence floated around along before then.
What would your first guess be
for the biggest sofa that could fit around this hallway?
What does it look like?
Let's think about it mathematically.
First of all
I should mention we are working in two dimensions.
We'll set the width of the hallway
to one meter since the exact value
is not important for solving the problem.
Your first guess
for the maximum area sofa might be a snugly fitting square
with an area of one meter squared
but we can also rotate the sofa, so maybe we can do better.
How about a semicircle?
This semicircle has a one meter radius taking our sofa
to an area of 1.57 square meters.
As the semi-circle is always touching the hallway walls
we can't make it any bigger.
Is this already our ultimate limit?
Surely we can push the boundaries further.
A guy named Hammersley had the idea to cut
the semi-circle in half, pull the pieces apart
and place a rectangle between them.
Then carve another semicircle out of the rectangle.
This sofa is larger than the semicircle.
If you do the math, you'll find
that it has a rather unusual area
of PI on two plus two on Pi
which is about 2.2074 meters squared.
A pretty big improvement.
So is this the biggest sofa that can fit around our hallway?
This still relatively simple shape is only bound
by straight lines and circles
and those shapes are often the solution
to simple optimization problems.
The shortest path between two points is a straight line
and a circle is the shape
with the largest area for a given circumference.
Mathematicians love these shapes
and they love elegant solutions.
So at first glance, it's intuitively appealing
that Hammersley sofa is the largest sofa
and that's what everyone thought for 24 years.
No one could find a larger sofa as hard as they tried.
That is until Joseph Gerver.
Gerver saw that by rounding
off the edges at the end of the inner semicircle,
you could add more area to the top of the sofa.
Gerver's sofa has an area of about 2.2195 meters squared,
0.5% larger than Hamersley's.
Admittedly, it's not a huge improvement.
And yes, it looks very similar to Hammersley sofa
but what's amazing about Gerver's sofa
is not how much bigger it is,
but about the way he came up with it.
The brilliance of his thinking was just phenomenal.
To understand his basic idea, we need to ask why
is this seemingly simple question so hard to answer
in the first place?
Well, there are two reasons.
So far We've just been guessing at shapes
and there are a lot of shapes out there, infinitely many.
In fact, even for someone with good geometric intuition
it's not always clear which shapes will and won't work.
Like out of all of these shapes,
can you guess which ones will make it around the corner?
Pause now, to make a guess.
(upbeat music)
So we're essentially using a trial
and error method on infinitely many possible shapes.
The second reason is a bit less obvious.
Let's go back to our good old square.
It uses only translational motion to fit around the corner.
That is, it moves purely in straight lines.
Our semicircle only uses rotational motion
to fit around the corner.
It rotates on its axis while staying in place.
Hammersley sofa uses a combination
of both translational and rotational motion.
Depending on the shape,
there are infinitely many combinations of translational
and rotational motion.
So out of infinitely many shapes,
taking infinitely many paths,
you need to find the one with the maximum area.
That's what makes the problem so difficult to solve.
So how did Gerver tackle these issues?
He thought about the problem from a different perspective.
Rather than moving a sofa through a hallway
he thought of moving a hallway around a sofa.
Here I've got some really nice interactive animations made
on Wolfram Mathematica.
I've posted the links to all of them in the description
if you wanna have a play around with them yourself.
So here we have the regular way to think
about the sofa problem,
a sofa moving around a stationary hallway.
Now this is how Gerver thought about it.
A hallway moving around a stationary sofa.
Here are the two perspectives side by side.
So far we've been coming up with the shape first
and then seeing the resulting path through the hallway.
Gerver flipped this.
He started with the path of the hallway
and then saw the resulting shape.
Look at all the different sofa shapes we get
by first choosing the path for the hallway to follow.
This is a pretty weird one.
How does this help us?
Well, because all of these shapes are created
by the movement of the hallway,
we know that they at least all fit around the corner.
We've eliminated all the sofas
that don't fit around the hallway.
Now all that's left to do is find the biggest one.
I say that like it's easy when really it looks like this.
It looks complicated.
But all of this math actually represents a simple
and brilliant insight, an insight that Gerver thought led
to the optimal path.
To understand, we first need to look at a very crappy sofa.
The reason it looks so crappy
is because we're only considering three times steps
of the hallway's rotation.
Imagine taking a photo at three different points
of the hallway as it rotates.
It's a crude approximation, but don't worry
its crudeness will help us demonstrate Gerver's brilliance.
Imagine I come up to you and say, "Here's the path
of the hallway that gives the largest sofa area."
If you were Gerver, you would say, "No, it's not."
And I'd say, "What? Yes it is."
And you'd say, "No, it's not."
And I'd say, "Okay, rude, prove it."
And you'd take this hallway
and slide it up in this direction
and say, "Look, this sofa has a larger area
than the one you showed me."
And I'd say, "How do you know that?"
And you'd say, "It's simple.
Go back to your inferior sofa.
The one you just claimed was the biggest.
Look at these two edges.
The one highlighted in green is longer
than the one highlighted in red, right?
So when I shift the hallway in this direction
I'm gaining more area at the top
of the sofa than what I'm losing at the bottom of the sofa.
Therefore, this sofa is bigger
than the one you just showed me.
Your path of the hallway
was not the one that gives the larger sofa."
And then I say, "Hmm, I see your point.
So how do we get around this?"
And you say, "Well, because I'm Gerver,
I have the brilliant insight
that the largest sofa must be the one
where all parallel sides are of equal length like this one."
This property where parallel sides
are equal is called balanced.
And I ask, "What does that do?"
And you say,
"Now we can't shift the hallways in a way
where we gain more area."
No matter how we tweak the hallway's path,
when the sofa shape is a balanced polygon
we always either lose more area than we gain
or lose the same amount of area as we gain.
So now to move from this crappy crude sofa
to Gerver's final sofa, we move from our three snapshots
of the hallway rotation back to the full hallway rotation.
If you wanna say it in calculus
we take the limit of infinitely many hallways.
To sum up, Gerver's final sofa
is the result of finding the path
of the hallway that gives a balanced sofa.
This ensures that small movements
in any direction will not increase its area.
Creative solutions like this, leave me in awe.
They are a big reason why I fell in love with mathematics.
It's the way of thinking that amazes me.
I'm always trying to improve my problem solving skills.
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Who knows?
Equipped with your advanced problem solving skills,
maybe you'll find a bigger sofa than Gerver's.
I said at the beginning
that the moving sofa problem remains unsolved.
And that's because Gerver could never prove
that his was the biggest sofa.
It's just the biggest one that anyone's ever found.
Gerver only showed that small changes
to his sofa can't increase the area.
A radically different looking sofa might be larger.
The Polish mathematician, Michael Batsch had a very
important insight regarding Gerver's sofa, which he wrote
in the beautiful last sentence of his 2022 publication.
"Not only is it highly likely to be the optimal solution
to the posed problem
but it is also possible to design a neat piece
of furniture offering space
for rather a large coffee table."
What I like about this problem is that it's hard to solve
but easy to understand and think about yourself.
You can consider many different variations.
Most of the current solutions
to these variations use similar methods to Gerver's
and are also conjectured to be the optimal solution
but not proven.
So I leave you with this important question.
Is there a larger sofa out there waiting to be discovered?
(upbeat music)