This open problem taught me what topology is

Here's a question that nobody in the world knows the answer to.

Suppose you have some closed continuous curve,

which essentially means some squiggle you could draw on paper without lifting the pen,

that ends where it starts.

If you can find four points somewhere on this loop that make the vertices of a square,

it's called an inscribed square of the loop.

And the unsolved question is whether every possible closed

continuous loop like this necessarily has an inscribed square.

The question was originally posed by Otto Toeplitz in 1911,

and it's commonly known as the inscribed square problem.

One of my all-time favorite pieces of math, easily in the top five,

is a very beautiful proof for a simpler version of this question,

asking instead whether you can necessarily find an inscribed rectangle.

The argument is originally due to Herbert Vaughan, and as a spoiler,

our discussion here is going to lead you to think about this shape right here,

what's known as a Klein bottle, but it won't come up as some curiosity or a trinket.

It arises as a natural problem-solving tool.

One of the earliest videos that I made for this channel was actually about this proof,

and I'm making this video as a kind of second edition to that one.

I realize second editions are much less of a thing for YouTube videos than

they are for books, but part of my motivation here is that there's been

more research since that original video that's very much worth discussing.

Another motivation is that there's numerous caveats and interesting connections

both beautiful and mind-broadening that are definitely worth including.

Aside from that, given that it's one of my favorite pieces of math,

I kind of just want to take a stab at animating it and motivating each one of the steps

as best as I know how.

Some of you might ask why on earth anyone would care about

proving that any closed loop has an inscribed rectangle.

I personally don't know of any application.

Frankly, I would be surprised if there was one.

But I think many of you know that engaging with challenging puzzles,

even when they're pure puzzles, has a way of sharpening your problem-solving

instincts in a way that can be carried over to other practical applications later.

But I can also give you a much more specific reason why I love this particular proof.

It's something I first saw a long time ago, and I remember it being the

first time that I felt a sense for what topology is actually all about.

You see, in a lot of recreational math settings,

topology is sometimes presented as something like a study of bizarre shapes.

A common classroom activity might be to apply a half-twist to a thin piece

of paper and glue the ends together, forming what's called a Möbius strip.

You might be told this is what's called a non-orientable surface.

Loosely speaking, this means there's no notion of an inner side or an outer side.

The whole thing just has one side.

Topology is also often presented as a sort of rubber sheet geometry,

where shapes are considered the same if you can deform one into another without

tearing it.

Neither of these notions, in my opinion, really captures what it's actually about.

And when I was growing up, I remember these kinds

of examples left a frustrating open question.

How is this math?

How does any of this actually help you solve problems?

If you stick with me to the end here, you'll see how these shapes and their bizarre

properties are not just idle curiosities, but they're actual tools for logic and

deduction.

The first step in proving that any closed loop contains a

rectangle is going to be to reframe the question just a little bit.

Instead of thinking about four points that are the vertices of a rectangle,

think about searching for two distinct pairs of points,

such that the lines connecting each pair have the same midpoint and they

have the same length.

Hopefully it's not too hard to convince yourself that this

really is the same thing as searching for a rectangle.

If I told you, hey, I found two line segments somewhere out in space,

and I further specify that both of them have the same center,

and also that both segments have the same length,

then the four endpoints of those two different lines have to form a rectangle.

If you want, you could try to pause and ponder to rigorously prove this.

It's a relatively straightforward geometry exercise.

Given some arbitrary closed loop, what you and I are going to do

is somehow think about all possible pairs of points on that loop.

For any one of these pairs, we care about two things.

The first is where its midpoint sits, which you

might think of as two numbers worth of data.

The xy coordinates on the plane where the loop sits.

The other thing we care about is the distance between those points,

which is another data point.

Now if you're a mathematician and you see three numbers worth of information like this,

it is a very natural step to try packaging them together and think of that

data as being a single point in a three-dimensional space.

In our example, if you imagine the loop sitting on an xy plane inside that space,

and its midpoint has some coordinates xy, then this 3d point that we care about,

the one packaging x, y, and d, could be thought about as a point directly above that

midpoint, such that the distance off the plane matches the distance between the pair of

points on the loop.

Some other pair of points on the loop would correspond

to some other point in three-dimensional space.

And in essence, what we have here is a mapping.

A mapping from pairs of points on the loop to three-dimensional space.

The important feature of this mapping that we're going to rely on is that it's continuous.

And essentially what this means is if you just slightly wiggle the input,

slightly nudging that pair of points, the output only slightly wiggles as well.

There are never any sudden jumps.