Then I add those two together to get 2358 What would happen if instead, I start with three numbers, and now I'm gonna add all three together.

So that gives me three.

And I'm now going to add these three together that's going to give me five, and then these three together that's going to give me nine on Dhe.

Then I'm going to get 95 on 3 17 and we're going to get a different sequence off numbers, or it's a bit like the seven Artery, but it's got free instead off, too.

So it's called the trip Banacci.

Let's call this on DSO.

This is the Tribune, our chief T end, and you have that f n plus one divided by F and goes to the golden ratio so we can also do the same thing with the Tribune Archie numbers, but they go to what's called the Tribune.

Archie Constant, with the Fibonacci numbers, won the ways we can sort of think about what?

How they're going to end up, Let's say, Well, let's start this one x x squared on X squared was equal to the two previous ones together.

So here this is X squared is equal to X plus one.

Here we're adding the three previous term.

So we get one x x squared.

This term is going to be the sum of all three.

So it's X cubed.

And now we have X cubed is equal to X squared plus X plus one.

And so that's a polynomial.

Whilst I can reasonably well solve quadratic ce que bix have a Farmall messy to salt.

So I'm gonna get a calculator to do that.

So this gives us 1.83929 That's gonna trip on Archie Constant Tribune.

Archie Constant.

I'm assuming there's a four Banacci constant and a Fibonacci called The End Been Archie's on dhe.

So here I've got interesting shape.

It's Scott er a fractal boundary going around it.

It's got a nice property too.

So I've got this style here on Dhe.

This is the same shape, but a little bit smaller on.

Then I've got again the same shape, but smaller the edges of these.

I confined it to get in the right way out on DDE that we got so the edges are such so This shape at different sizes actually fits onto itself.

And that means that these edges might must either be straight lines, which they clearly not or fractal.

It's not a fractal itself, but the boundary is a fractal.

So now I will take 1/3 shape and put them there on dhe magically, I have my same shape.

These are three sizes off this one shape.

And when I put those three together, I get the same.

If this one scales up to this one, this one scales up to this one and then this one scales up Thio the whole thing.

I can take this one step.

So what I've done here is make a larger version off this object.

So this piece here becomes this tile here.

This piece here becomes this tile here and then this tile here is replaced by the whole piece hits the the three tiles.

If I rotate this there, you see that this is just these three types.

So now we could imagine during this another time In another time we can get larger and larger regions.

We could ask for how many tiles I am.

I gonna have in them.

Well, this Patton is made up off one of the smallest tiles, one of the medium tiles on dhe one off the large tiles.

This is made up off one medium, one large on, then one of the scale size big on large, which itself is made up off one small, one medium in one lunch on.

When I go into the next step up, I'm gonna have one copy of this.

One copy of this on, then.

One additional lunch.

So each stage I'm going to take the three previous pieces, put them together to get the next one up.

And so what are we going to have?

It's back over here.

I have my won.

Won won.

I put those together, so I get 31 one on three.

Give me my five.

The next one up is going to be my five.

My three on my one, which is going to give me nine and so on.

So this shape is called the Drowsy Fractal, and it was found by Gerald Rousey.

The interest is sort of studying how in matrices act on three dimensional space and there's a lot of nice theory for help for certain properties of how matrices act on two dimensional space, especially a matrix, which has one expansion on one contraction.

What that will do is you have these things got Eigen vectors and you pull down in one direction and expand.

Everything gets squashed down and pulled out.

There's a whole theory in that cord.

Markoff partitions, which how basically have nice properties as you get pulled down and smeared out.

You couldn't turn that into a nice periodic what?

Rousey asked.

Well, what about for the cubic case when we're dealing with three dimensional space?

So the specific matrix you're studying here has as one of its Eigen values this number, the Tribune actually constant.

And so because the Tribune archery constant is turning up in the thing you're trying to study, that's how it cut turns out in these.

These tiles and the golden ratio of Fibonacci numbers are really important in the study off the two dimensional Markoff partitions.

And so when you go up to three dimensions, you have to deal with cubic things.

It's quite natural.

It's sort of almost the ancient Greeks who will felt that a cubic number really waas a cube.

So they didn't.

Some people say they didn't even believe in fourth powers.

When you're dealing with three dimensions, you naturally have cubic stuff.

Do they exist in some flowers and things that they don't exist in nature in the way that they just like the Fibonacci, is quite as popular by nature's is meant to be on The only place beyond this that I know of them is an object called the Snub Cube, which is quite an attractive little Polly Hedren.

Hey there, everyone.

Just a reminder that there's now a number file pod.

Cursed.

Have you had a listen yet?

This is what it looks like on your podcast.

APS just search for number file.

And if you haven't got into this whole podcasting thing that you still wanna have a listen, I am putting the episodes on the number file to YouTube channel.