So a frieze pattern is a table of numbers. The rule of the game is we start with a row of 1s.

Then we'll have more interesting rows, but eventually we want to end up with a row of 1s again.

So I will do that.

Okay, so there is one parameter in what we are doing:

the number of rows, 1, 2, 3, 4, here.

So there are, say, 4 rows.

And now I will put some numbers in the first row, and I will start, fill out this table.

There is only one rule. The rule is that whenever you see a small diamond like this,

the product of these two numbers minus the product of these two numbers must be always 1.

WE-NS is always 1. That's the rule.

Okay, what I'll do, I'll put some "random" numbers. Of course, they are not random,

but at this point you are not supposed to understand what they are.

In the first row, and actually the row is infinite,

so I put the seven numbers and then I repeat them. So, it's 7-periodic.

Okay, so it's, should do 1,3, and so on. So now we are in business.

Using this diamond rule, we can start to fill out this table. I will do it myself, but, Brady please watch me.

Let's see what should be here. The number here times 1 should be 1 less than this product,

so 3 minus 1, this should be 2. Let's see. 1 times 3 minus 1 times 2 is 1.

Brady: Perfect. Sergei: Next one. Here, 6 minus 1, 5, this should be 5.

3 times 2, 6. 1 times 5, 5. 6 minus 5, 1. Correct?

Okay, here. 2 times 2, 4. I put 3 here. Checking: 2 times 2, 4, minus 1 times 3, 3, 4 minus 3, 1.

Brady: Okay. Sergei: Okay, here, 1. 2 minus 1.

Here, 3. 4 minus 3.

Here, well this is a big one, 7. 4 times 2, 8, minus 7, 1.

Here, 2, 1, 1, 1.

1, 2, 3, 4, 5, 6, 7.

This will also repeat because of the first row repeats with period 7, so the next one should be 2.

Brady: So, so far It's been good. We got nice, we got integers, no fractions, nothing complicated.

Sergei: Yes. I want to point out that a small miracle has already happened.

So if look at this rule, this compass rule,

If you know three of these numbers, you can always solve for the remaining one.

For example, suppose you know W, E, and N, and you want to solve for S.

So what is S? S is WE-1 over N. We subtract, and we divide.

So there is no chance, in general, that will stay within positive integers.

We divide, so in principle, these should be fractions.

Yeah. What we should expect in general if we put rational numbers in the first row,

Everything inside should be rational. But I put positive integers and so far it's stable in the positive integers.

That's already strange.

Alright. So let's do hard work. What should we put here. 10 minus 1 divided by 3, 3. 10 minus 9.

What should we put here, 15 minus 1 over 2. 7.

Here, 3 minus 1. That's 1. Here, 2.

Here, 21 minus 1 over 4. 5.

7 minus 1, 6 over 2. 3.

Here, 1. 1, 2, 3, 4, 5, 6, 7. It should be periodic, so the next one must be 3.

Brady: The miracle continues! Sergei: The miracle continues, you are right.

And, you want the surprise that it will continue to the end.

So two more miracles we expect. We expect positive integers in this row, and we expect this one to satisfy the compass rule.

Okay, so let me do it a little faster.

That will be 4. That will be 2. That will be 1. That will be 3.

2, I guess?

2 and 1. And it should be periodic so the previous one just like this one.

And finally, I hope, it all works. 4 minus 3, 1, 8 minus 7, 1,

1, 1, 1, 1, 1, 1. Yes.

Brady: So, you filled the sandwich perfectly and you only used integers.

Sergei: Right, so you start to wonder what's going on. Brady: Did you choose magic numbers at the start?

Well, certainly, certainly the numbers were not random. If you change anything at all randomly, it will break completely.

The secret has a name. It's a theorem.

It's due to two famous mathematicians, one of them unfortunately not with us, another alive.

Coxeter and Conway. John Conway, who is a character in your movies.

John: "I'm not gonna worry anymore, ever again."

John: I was going to study whatever I thought was interesting.

So the theorem explains this phenomenon, and actually gives a complete description, complete classification,

of frieze patterns which consist of positive integers.

So again, I remind everyone that there is a parameter here, the number of non-trivial rows.

Here, this number is 4, but in principle, can be handwritten in anything else.

So the theorem is not one theorem, but infinitely many theorems, one for each parameter.

To explain what's going on, I need to draw a polygon which will have seven sides.

I will partition this heptagon into triangles. I will triangulate it by its diagonals.

I will draw this diagonal,

and this diagonal,

and maybe this one,

and possibly this one.

Okay, so now we have five triangles which make this heptagon,

and I will write numbers at every vertex, and the number is the number of adjacent triangles.

So, for example, this vertex has exactly one triangle adjacent to it, so I write 1 here.

This vertex has four triangles, so that is 4.

This one has 2. 1, 3, 2, and 2. Okay, now we have seven numbers.

If you examine this first non-trivial row, you will recognize these numbers starting with this 1, I guess.

1, 3, 2, 2, 1, 4, 2.

And then we repeat it. Okay? Brady: So this shape gave you your numbers?

Sergei: Yes, exactly.

And the brilliant, beautiful theorem of Coxeter and Conway says that this miracle will happen every time.

Given an n-gon, we triangulate by diagonals any way we want,

it's an interesting question how many ways, I will say something about that,

and we put the numbers of triangles adjacent to every vertex around the polygon.

And now we create the first seed row, which will be n-periodic. Here it was, n was seven.

And we put these numbers in the row, with period n, and then we start to fill out our table using the compass rule.

So the claim is that after n minus 3 non-trivial rows, which all consist of positive integers,

we will again have a row of 1s. Brady: So that's when you get back to the 1s.

Sergei: Exactly. Brady: n minus 3.

Sergei: n minus 3. Yes.

So that's the way to construct a frieze pattern consisting of positive integers of width n minus 3,

and the theorem is that they all are obtained this way, so it's a one-to-one correspondence.

If we start with the square this gives us period 4 in the horizontal direction and width 1.

Pentagon gives us a 5-periodic pattern with width 2.

Hexagon, 6-periodic, width 3.

Hep-heptagon you already saw, but you can go to higher, well probably there is not enough room for, here,

But it works for every n. Maybe we should start with a square.

There are exactly two ways to triangulate it by diagonals. This is one way, and this is another way,

and although one is obtained from another by rotation, we consider them as different,

so there should be two different frieze patterns of period 4 and width 1.

According to these numbers, which are 1, 2, 1, 2, and 2, 1, 2, 1.

Not very interesting, but it works. Reality check.

Pentagon has five different triangulations by diagonals.

They all look the same because the numbers which we write form the same sequence, 3, 1, 2, 2, 1,

but again, we consider these triangulations as different, so there are 5 ways to do it.

So, 2 ways here, 5 ways here, maybe for complete record we should put triangle which has only 1 triangulation,

which you don't need to do, it's already there, and the numbers of course are 1, 1, 1.

Hexagon. Hexagon really takes some work. So, this is one.

This is a different kind of pattern. I guess there is one other pattern, like so.

Brady: That's gonna start giving us quite different numbers isn't it? Sergei: That is correct.

If you are willing to believe me, the number will be 14 here, and actually, this sequence, 1, 2, 5, 14,

this is a famous sequence, which you can find everywhere. These are Catalan numbers,

and they are all over mathematics, all over combinatorics.

There are many, many, many interpretations, This is just one of the most common ones.

Sure, the next one I guess is 42. Brady: And this is a heptagon.

Sergei: This should be heptagon. Yes, indeed.

Brady: So does that mean as our polygons get bigger, they'll give us different families of our frieze seeding numbers?

Sergei: That is correct. Yes. The first row will depend on n as an n-gon, so it will be n periodic,

and on the triangulation. Different triangulations give us different seed rows.

Brady: So I want to make a 4-rowed sandwich here, Sergei: Ok.

Brady: one of these 4 row sandwiches, obviously I have to use a heptagon, Sergei: Correct.

Brady: but I will have different ways to seed that depending on how I triangulated my heptagon. Sergei: Exactly right. Exactly right.

So in this example, we chose one particular triangulation, but, for example, you can choose one of the diagonals,

let's say this one, erase it, and replace, for example by this one, and that would be a different frieze pattern,

different triangulation, the numbers will change as well.

Brady: This will always work, no- for every triangulation it will always give me frieze numbers that will...

Sergei: This will always works, work, and this is the only thing which will work.

So the theorem is that if you want a frieze pattern consisting of positive integers, you should do this, and they all will be obtained this way.

Brady: So there's no chance that I could devise a sequence that won't come from a triangulation.

Sergei: Absolutely not. Absolutely not. You don't want to see the proof, but it's a theorem,

so it has a proof, it's 100% certain, yes. Brady: But it's a, this is... amazing.

Sergei: It's amazing. But you, it takes two geniuses, you know. Coxeter and Conway.

It's not just any random pair of mathematicians. Yeah, it's really beautiful.

Maybe I should say a few words about the place the subject occupies in contemporary mathematics.

So, when Coxeter and Conway came up with their result, and when Conway introduced frieze patterns,

it seemed like a marginal subject in some sense.

I mean there are many beautiful things in mathematics which do not belong to mainstream.

Fortunately for the subject, fortunately for all us, now it's really mainstream.

So there is a new, relatively new, a theory called the theory of cluster algebras, and this is about 20 years old,

and these frieze patterns occupy a very honorable position in this theory, it's one of the main examples,

so they're very much studied nowadays, and they are very popular.

If you google the subject you will see many, many scientific papers on this subject.

So, the intuition of Coxeter and Conway proved itself excellent. They recognized the beauty and now it's important, too.

Brady: To see even more from this interview, including the famed lightning bolt in frieze patterns,

have a look at our extra footage channel, Numberphile2. There'll be a link on the screen and in the description.

This video was filmed at the Mathematical Sciences Research Institute, or MSRI.

If you'd like to find out more about MSRI, have a listen to our podcast episode with the director of the institute, David Eisenbud.

Again, links are on the screen and down in the description.