What's so special about the Mandelbrot Set? - Numberphile

Maybe this is "Mandelbrot set back to basics". I need your help though, can you think of a number?

(Brady: 7) - Good. Can you square that number?

- (49) - Can you do this while filming? You've done it correctly so far, okay, next question you're gonna get upset - square 49.

- (Nup)

Okay, good answer. The thing is I don't really care about the answer. What I'm caring about is an iteration.

I'm gonna keep asking you to do the same thing again and again

and if I ask you to pick a number and keep squaring it, you already know what's gonna happen, right?

It's gonna get big. - (Brady: It'll blow up.) - Is it? Square 1.

- (Okay, except 1.) - Okay good, now square a half.

(Ok, it's gonna get smaller)

So sometimes it blows up, and actually that happens when it's bigger than 1, and it becomes quite obvious when you try it.

But 1 doesn't blow up, so 1's different. And numbers less than 1, like 1/2,

actually become a quarter when you square them and then become a sixteenth and so on. But then numbers much less than one like minus

six...

and that one's blown up as well. So what I'm really talking about, and what iteration is quite often talking about, is when iterations are stable,

when they sort of head towards somewhere you can see, and when they're not. And it's best to see this on a diagram,

so have a look at my screen here.

You picked the number seven, which is kind of off my screen here, but these red arrows, I'm moving them around.

This is not the greatest diagram, let me get this clear in this to start with, it's just a line.

But the arrows are heading that way because when I square this number

I'm moving around they go that way and they never come back, much like your seven did.

But if I go less than this number one, they don't go that way anymore, they go that way.

In fact, they head towards zero and they're stable there.

So for the sake of moving it around, it's kind of nice to see the difference, it's a really sharp line

then it completely changes behaviour, but that way they're unstable, that way they're stable. Stable, unstable, stable, unstable,

stable, stable, stable unstable - you get the idea.

In fact if you go negative

they're still stable. The head positive because when you square a number that's negative you get positive.

Still head toward zero until you hit minus 1 and then

unstable. Stable, stable, unstable, stable. Now despite all my commentary

it's not that exciting, yet. And if you know anything about the theory of

iterations and the consequences of them, or if you've watched any other video about the Mandelbrot set you know that

we're not just talking about real numbers, i.e the numbers on the line

I've just been showing you, we're talking about numbers in two dimensions. So

complex numbers. And I'm not gonna

give you an introduction to complex numbers here because I don't think you need to hear it and there are other places you can get

them.

Holly on the Numberphile Channel has done excellent videos about how this happens,

but I want to show you the same comments that she's made on brown paper. I'd like to show you

moving. Here's my two-dimensional number line. Here is the number that you get to pick randomly. We can still do it in one dimension

just by sliding along here and the other dot that's moving around this tiny one is the square of the other dot.

So 1 squared is about 1, bigger than 1 it heads off

that way, lower than 1 it goes that way; but I can move off the line now. And it has a weirdly circular flavour to it,

which I kind of like. It's like

chasing the number around, it moves quicker than you expect and that's already quite pleasing if you're a nerd like me.

But what I'm really interested is the iteration if you keep doing the instruction: square it, square it, square it.

We saw what happened on the real line. In two dimensions it looks different. So it looks like this.

So this line indicates if we start there it goes to there when you square it and then to there and there. And so if