Name: Frieze Patterns - Numberphile
Uploaded: 2020-03-31T07:20:39.000Z
Duration: 13 min
Description: Thousands of YouTube videos with English-Chinese subtitles! Now you can learn to understand native speakers, expand your vocabulary, and improve your pronunciation...

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

Adverbs of manner

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

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

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.

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

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.

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, 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.

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

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,

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.

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.

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.

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

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.

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.

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,

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Frieze Patterns - Numberphile

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