Who cares about topology? (Old version)

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 I jump into it, it's worth saying a few words on why I'm excited to share this solution. When I was a kid, since I loved math and sought out various mathy things,

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,

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,

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 help to solve problems? It wasn't until I saw the problem that I'm about to show you,

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 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

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.

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.

no matter how crazy, has at least one inscribed square. Pretty interesting, right? , 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 inscribed square in it. Now, if we weaken the question a bit and ask about inscribed rectangles

Now, if we weaken the question a bit and ask about inscribed rectangles instead of inscribed squares, it's still pretty hard, but there is a beautiful, video-worthy solution that might 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.

points on the loop and instead onto pairs of points. We'll use the following fact about rectangles. Let's label the vertices of some rectangle ABCD. Then the pair of points AC has a few things in common with the pair of points BD. The distance between A and C equals the distance between B and 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, any time you have two separate pairs of points in space, AC and BD, if you can guarantee that they share a midpoint and that the distance between AC 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.

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,

in pairs of points on the loop and outputs a single point in 3D space, which encodes the midpoint and distance information. It will be like a graph. Consider the closed loop to be sitting on the xy-plane in 3D space. For a given pair of points, label their midpoint m,

For a given pair of points, label their 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,

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 surface above the plane. Now look at the surface, and notice how it seems to hug the loop itself. This is going to be important later, so let's think about why it happens.

This is 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.