I've got several fun things for you this video.
An unsolved problem, a very elegant solution to a weaker version of the problem,
and a little bit about what topology is, and why people care.
But before we jump into it,
it's worth saying a few words on
why I'm excited to share the solution.
When I was a kid since I loved math and sought out various mathy things,
I would occasionally find myself in some talk or a seminar
where people wanted to get the youth excited about things
that mathematicians care about.
A very common go-to topic to excite our imaginations was "topology".
We might be shown something like a Mobius strip,
maybe building it out of construction paper by twisting a rectangle and gluing its ends.
"Look!", we'd be told as we were asked to draw a line along the surface.
"It's a surface with just one side!"
Or we might be told that topologists view coffee mugs and donuts as the same thing
since each has just one hole.
But these kinds of demos always left a lurking question.
"How is this math?"
"How does any of this actually help to solve problems?"
it wasn't until I saw the problem that I'm about to show you
with its elegant and surprising solution
that I started to understand why mathematicians actually care about
some of these shapes and the properties they have.
So there's this unsolved problem
called "the inscribed square problem".
If you have a closed-loop
meaning you squiggle some line through space in a potentially crazy way
and you end up back where you started.
The question is whether or not you'll always be able to find four points on this loop
that make up a square.
If your closed loop was a circle, for example,
it's quite easy to find an inscribed square.
Infinitely many, in fact.
If your loop was, instead, an ellipse,
it's still pretty easy to find an inscribed square.
The question is whether or not every possible closed-loop,
no matter how crazy, has at least one inscribed square.
Pretty interesting, right?
I mean, just the fact that this is unsolved is interesting
that the current tools of math can neither confirm nor deny
that there's some loop with no inscribe square in it.
Now, if we weaken the question of it
and ask about inscribed rectangles
instead of inscribed squares
it's still pretty hard.
But there is a beautiful video-worthy solution
that might actually be my favorite piece of math.
The idea is to shift the focus away from individual points on the loop
and, instead, onto pairs of points.
We'll use the following facts about rectangles.
Let's label the vertices of some rectangle a, b, c, d.
Then the pair of points a, c has a few things in common with the pair of points b, d.
The distance between a and c equals the distance between b and d
and the midpoint of a and c is the same as the midpoint of b and d.
In fact, anytime you have two separate pairs of points in space a, c and b, d
if you can guarantee that they share a midpoint
and that the distance between a, c equals the distance between b and d
it's enough to guarantee that those four points make up a rectangle.
So what we're going to do
is try to prove that for any closed loop
it's always possible to find two distinct pairs of points on that loop
that share a midpoint and which are the same distance apart.
Take a moment to make sure that's clear.
We're finding two distinct pairs of points
that share a common midpoint and which are the same distance apart.
The way we'll go about this
is to define a function that takes in pairs of points on the loop
and outputs a single point in 3d space
which kind of encodes the midpoint and distance information.
It will be sort of like a graph.
Consider the closed-loop to be sitting on the xy plane in 3d space.
For a given pair of points, label the midpoint M
which will be some point on the xy plane
and label the distance between them, d.
Plot the point which is exactly d units above that midpoint M
in the z direction.
As you do this for many possible pairs of points
you'll effectively be drawing through 3d space
and if you do it for all possible pairs of points on the loop
you'll draw out some kind of surface above the plane.
Now look at the surface
and notice how it seems to hug the loop itself.
This is actually going to be important later.
So let's think about why it happens.
As the pair of points on the loop gets closer and closer
the plotted point gets lower
since its height is, by definition, equal to the distance between the points.
Also the midpoint gets closer and closer to the loop
as the points approach each other.
Once the pair of points coincides
meaning the input of our function looks like (X, X) for some point X on the loop
the plotted point of the surface will be exactly on the loop at the point X.
OK, so remember that.
Another important fact
is that this function is continuous
and all that really means is that if you slightly adjust a given pair of points
then the corresponding output in 3d space
is only slightly adjusted as well.
There's never a sudden discontinuous jump.
Our goal, then, is to show that this function has a collision.
The two distinct pairs of points
each map to the same spot in 3d space.
Because the only way for that to happen
is if they share a common midpoint
and if their distance d apart from each other is the same.
So in some sense, finding an inscribed rectangle
comes down to showing that this surface has to intersect itself.
To move forward from here
we need to build up a relationship with the idea of pairs of points on a loop.
Think about how we represent pairs of real numbers
using a two-dimensional coordinate plane.
Analogous to this, we're going to seek out a certain 2d surface
which naturally represents all pairs of points on the loop.
Understanding the properties of this surface
will help to show why the graph that we just defined has to intersect itself.
Now, when I say a pair of points
there are two things that I could be talking about.
The first is "ordered" pairs of points
which would mean a pair like (a, b)
would be considered distinct from the pair (b, a).
That is there some notion of which point is the first one.
The second idea is "unordered" points
where {a, b} and {b, a} would be considered the same thing,
where all that really matters is what the points are
and there's no meaning to which one is first.
Ultimately, we want to understand “unordered” pairs of points.
But to get there, we need to take a path of thought through “ordered” pairs.
We'll start out by straightening out the loop
cutting it at some point and deforming it into an interval.
For the sake of having some labels
let's say that this is the interval on the number line from 0 to 1.
By following where each point ends up,
every point on the loop corresponds with a unique number on this interval.
Except for the point where the cut happened
which corresponds simultaneously to both endpoints of the interval
meaning the number 0 and 1.
Now the benefit of straightening out this loop like this
is that we can start thinking about pairs of points
the same way we think about pairs of numbers.
Make a y-axis using a second interval
then associate each pair of values on the interval with a single point
in this 1x1 square that they span out.
Every individual point of this square
naturally corresponds to a pair of points on the loop
since its x and y coordinates are each numbers between 0 and 1,
which are in turn associated to some unique point on the loop.
Remember, we're trying to find a surface
that naturally represents the set of all pairs of points on the loop
and this square is the first step to doing that.
The problem is that there's some ambiguity
when it comes to the edges of the square.
Remember, the endpoints 0 and 1 on the interval
really correspond to the same point of the loop
as if to say that those endpoints need to be glued together
if we're going to faithfully map back to the loop.
So, all of the points on the left edge of the square
like (0, 0.1), (0, 0.2) on and on and on
really represent the same pair of points on the loop
as the corresponding coordinates on the right edge of the square.
(1, 0.1), (1, 0.2) on and on and on.
So for this square to represent the pairs of points on the loop in a unique way
we need to glue this left edge to the right edge.
I'll mark each edge with some arrows
to remember how the edges need to be lined up.
Likewise, the bottom edge needs to be glued to the top edge
since y coordinates of 0 and 1 really represent the same second point
in a given pair of points on the loop.
If you bend the square to perform the gluing,
first rolling it into a cylinder to glue the left and right edges
then gluing the ends of that cylinder
which represent the top and bottom edges
we get a "torus", better known as the surface of a donut.
Every individual point on this torus
corresponds to a unique pair of points on the loop.
And likewise, every pair of points on the loop
corresponds to some unique point on this torus.
The torus is to pairs of points on the loop
what the xy plane is to pairs of points on the real number line.
The key property of this association
is that it's continuous both ways
meaning if you nudge any point on the torus by just a tiny amount
it corresponds to only a very slight nudge to the pair of points on the loop
and vice versa.
So if the torus is the natural shape for ordered pairs of points on the loop,
what's the natural shape for unordered pairs?
After all, the whole reason we're doing this
is to show the two distinct pairs of points on the loop
share a midpoint and are the same distance apart.
But if we consider a pair (a, b) to be distinct from (b, a)
then that would trivially give us two separate pairs
which have the same midpoint and distance apart.
That's like saying you can always find a rectangle
so long as you consider any pair of points to be a rectangle.
Not helpful!
So let's think about this.
Let's think about how to represent unordered pairs of points.
looking back at our unit square.
We need to say that the coordinates (0.2, 0.3)
represent the same pair as (0.3, 0.2)
or the (0.5, 0.7) really represents the same thing as (0.7, 0.5)
and in general any coordinates (x, y) has to represent the same thing as (y, x).
Once again, we capture this idea by gluing points together
when they're supposed to represent the same pair.
Which, in this case, requires folding the square over diagonally.
Now notice this diagonal line, the crease of the fold
it represents all pairs of points that look like (x, x)
meaning the pairs which are really just a single point written twice.
Right now, it's marked with a red line
and you should remember it
it will become important to know where all of these pairs like (x, x) live.
But we still have some arrows to glue together here.
We need to glue that bottom edge to the right edge
and the orientation with which we do this
is going to be important.
Points towards the left of the bottom edge
have to be glued to points towards the bottom of the right edge.
And points towards the right of the bottom edge
have to be glued to points towards the top of the right edge.
It's weird to think about, right?
Go ahead.
Pause and ponder this for a moment.
The trick which is kind of clever
is to make a diagonal cut
which we need to remember to glue back in just a moment.
After that, we can glue the bottom and the right like so.
But notice the orientation of the arrows here.
To glue back what we just cut
we don't simply connect the edges of this rectangle to get a cylinder.
We have to make a "twist".
Doing this in 3d space
the shape we get is a "Mobius strip"!
Isn't that awesome?
Evidently the surface which represents
all pairs of unordered points on the loop is the Mobius strip!
And notice the edge of the strip shown here in red
represents the pairs of points that look like (x, x),
those which are really just a single point listed twice.
The Mobius strip is to unordered pairs of points on the loop
what the xy plane is to pairs of real numbers.
That totally blew my mind when I first saw it!
Now, with this fact
that there is a continuous one-to-one association
between unordered pairs of points on the loop
and individual points on this Mobius strip,
we can solve the inscribed rectangle problem.
Remember, we had defined the special kind of graph in 3d space
where the loop was sitting in the xy plane.
For each pair of points
you consider their midpoint M which lives on the xy plane
and their distance d apart
and you plot a point which is exactly d units above M.
Because of the continuous one-to-one association between pairs of points on the loop
and the Mobius strip
this gives us a natural map
from the Mobius strip onto this surface in 3d space.
For every point on the Mobius strip,
consider the pair of points on the loop that it represents
then plug that pair of points into the special function.
And here's the key point.
When pairs of points on the loop are extremely close together
the output of the function is right above the loop itself
and in the extreme case of pairs of points like (x, x)
the output of the function is exactly on the loop
since points on this red edge of the Mobius strip
correspond to pairs like (x, x).
When the Mobius strip is mapped onto the surface
it must be done in such a way
that the edge of the strip gets mapped right onto that loop in the xy plane.
But if you stand back and think about it for a moment
considering the strange shape of the Mobius strip
there is no way to glue its edge to something two-dimensional
without forcing the strip to intersect itself.
Since points of the Mobius strip represent pairs of points on the loop.
If the strip intersect itself during this mapping
it means that there are at least two distinct pairs of points
that correspond to the same output on this surface.
Which means they share a midpoint
and are the same distance apart.
Which, in turn, means that they form a rectangle.
And that's the proof!
Or at least if you're willing to trust me
and saying that you can't glue the edge of a Mobius strip to a plane
without forcing it to intersect itself.
Then that's the proof!
This fact is intuitively clear
looking at the Mobius strip.
But in order to make it rigorous
you basically need to start developing the field of topology.
In fact, for any of you who have a topology class in your future
going through the exercise of trying to justify this
is a good way to gain an appreciation
for why topologists choose to make certain definitions
and I want you to take note of something here.
The reason for talking about the torus and the Mobius strip
was not because we were playing around with construction paper
or because we were daydreaming about deforming a coffee mug.
They came up as a natural way to understand pairs of points on a loop
and that's something that we needed to solve a concrete problem.
Alright, I have one final animation for you all.
But first I'd like to tell you a little about
what's making this and future videos possible.
First, I want to say a huge thanks to everybody who supported on Patreon.
I launched this only last week
and have been absolutely blown away
by people's enthusiasm for helping make more of these videos.
Right now, I'm working on an "Essence of Calculus" series
and those supporting on Patreon are getting early access to those videos
before I publish the full set in a few months.
I also want to thank "hover.com" for supporting this video.
I'm sure a lot of you watching have some idea for a website that you want to start.
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If you go there now and use the promo code "TOPOLOGY"
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That also lets them know that you came from me
which encourages them to support more videos like this one
so I can keep making them.
And with that, here's the final animation I promised.
It shows how that beautiful surface that we defined earlier
changes while the loop changes.