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#### 曼德博集合

The Mandelbrot Set - Numberphile

So, today I want to talk about the Mandelbrot set.

But I want to—

so there’s so many videos and websites and Java applets

and all of these things where you can see the beauty of the Mandelbrot set.

And this really nice fractal picture

and you can zoom in and see all of the interesting things.

[Music]
［音乐］
What I want to talk about is: What is this object?

So, what is this picture a picture of ?

Why do we care about this picture

other than just its intrinsic, sort of, appeal?

And, so just to generally talk about the way that

maybe mathematicians will look at the Mandelbrot set

[Music]
［音乐］
The first thing we need to understand is

that this entire thing is happening

in the world of complex numbers, okay?

So, if you remember, complex numbers and the complex plane,

the way that we view this is

that we have two axes

and we plot on this plane numbers of the form,

say a+bi,

and here these two things are real numbers

and’i’ is a symbol that means that i²=-1, okay?
i是一个满足i²=-1的符号
So most people are familiar with this but just a reminder.

It’s a convenience in some sense,

but there’s also a lot of useful information in this representation.

So for example, one thing that’s very natural

to look at is if I plot some complex number

say a+bi，

Okay, maybe this is something like you know 3+2i, something like that,

then one sort of natural quantity associated with this thing

is the distance from this point to the centre point of the plane.

And so this distance,

which we call the magnitude of the complex number,

it really has some inherent mathematical properties

that we really care about

and so the fact that this is so easy to visualize in the complex plane

you can also visualize like addition of complex numbers and subtraction and so on

in this plane in a very geometric way is really helpful.

[Music]
[音乐]
So how do we get to the Mandelbrot set from here?

So here’s sort of just the naive definition.

Let’s take a complex number C

and let’s associate to this complex number the following function:

so this is a function which takes as an input some complex number Z

and outputs Z squared plus C

So I’m thinking of this complex number

as being associated to this function,

and we what we’re interested in is

the behavior of 0 under iteration
0在迭代下的特性
So by iteration of fc(Z)

I mean what happens when I take 0

and I plug it into this function,

and then I keep doing that to the result.

So, for example, if we’re looking at f₁(Z)

Well, f₁(0)=0+1, which is 1 f₁(1),
f₁(0)=0+1=1 再算f₁(1)
so now I apply it to the answer that I got, right?

This is 1+1 which is 2.

f₁ of the previous thing

which was 2 is 2²+1 which is 5

f₁(5) is 5²+1 which is 26, and so on.

So that’s what I mean about

the behavior of 0 under iteration for a particular value of C.
0在特定C值下迭代的特性
Now what the Mandelbrot set is concerned with

is what happens to the size of these numbers

and by size I mean exactly what we were talking about before

about the distance from the number in the complex plane to this point 0, okay?

So it turns out there are two options for a function

fc(Z), defined to be Z²+C:

The first option is that the distance from 0

of the sequence we get, gets arbitrarily large.

BRADY: That means it blows up. DR. KRIEGER: It means it blows up.
– 那就是说它会“爆炸” – 就是说它会“爆炸”
It gets as large as you want it to be, okay?

So, this is what people mean when they say that the iterates go to infinity, okay?

They mean not necessarily that, okay,

they look like real numbers or integers or something like this,

but that the”size” of the number,in this sense, goes to infinity.

The other thing that can happen is that the distance is bounded.

The size is bounded.

So, and in fact you can say

that it never gets larger than 2.

So you have this sort of dichotomy

where only one of two things can happen

If you give me a complex number C

and I start iterating zero under that function Z²+C

either the distance of the iterates to 0 in this complex plane

gets really large for all of them,

so you can’t bounce back and forth, right?

It gets really large for all of them.

Or, it stays close to 0,

within a distance of 2 from 0.

So, for example, to illustrate these two cases,

we wrote down already a few iterates, under Z²+1 of 0,

and as you can see their size is growing

and in particular we’ve got some things that are further from 0 than 2 is,

and so this C=1 is case 1.

But there’s another possibility so let’s look at, well,

a good contrast maybe would be Z²-1.
Z²-1或许会与之形成鲜明对比
although this might be a little misleading

so if we look at, say 0,

and we start applying this function

Well, f_-1(0), that’s 0-1 which is -1.
f_-1(0)=0-1=-1
If we plug in -1 into that function

we have (-1)² which is 1, which is 0.

Oh, wait, okay, but we know what happens to 0, right?

It goes back to -1.

So these iterates just alternate between -1 and 0.

And so in particular they never get large, right?

So that’s an example of case 2.

So,the definition of the Mandelbrot set, then,

one definition of the Mandelbrot set,

which we usually call M, is the set of C,

complex numbers C, for which case 2 holds.

And I’m kind of all over the place here so let’s be clear,

Case 2

So,in other words,

if I look at the function represented by this complex number,

if I look at Z²+C and I start iterating 0 under that function,

everything remains bounded.

– It’s the guys that don’t blow up rather than the ones that do.
– 它不会像第一种情况一样“爆炸”
– That’s right,
– 没错
it’s the guys that don’t blow up instead of the ones that do.

And this is also

in case you’re curious how these pictures are always generated

So if you wanna figure out to draw a picture

whether C is in the Mandelbrot set or not.
C是否在曼德博集合里
Well, you just start iterating 0 under Z²+C

And if it takes a long time to get big

then you can give it one color

If it gets big really quickly you can give it a different color

and that’s how you get these shadings.

[Music]
［音乐］
I’ll point out here

that everything that’s in the Mandelbrot set

has to be within distance 2 of the centre, right?

Because of exactly this case 2 thing that I said

that once your iterate is larger than 2,

you’re out of the picture.

So the inside of this thing, let’s fill this in here,

this is what’s known as the Mandelbrot set

So let’s look at our examples, right?

So we had two examples, we had C=1

and we had C=-1.

So -1, is right here,
-1在这里
is indeed inside the Mandelbrot set.

1 is right here, and it’s outside.

let me take the easiest example inside of there

so we look at 0, right? C=0,

And we start iterating,well

what is the function associated to C=0?

f₀(Z) is Z²,

okay, so, let’s start iterating 0,

well, 0²=0
0²=0
So no matter how many times we apply

the function we just stay at 0.

– So you’re in the club. – So you’re in the club, that’s right.
– 所以0在曼德博集合里 – 所以0在曼德博集合里 没错
But if we take, say, some small number here

It’s a little hard to compute without taking a real number,

so I apologize,

but if we we take something like 1/8, something like that

if we start iterating so the first iterate is 1/8

and then you start adding things under iteration

but, but it’s never enough to get you

outside of that disc of radius 2.

– These guys are blowing up. – That’s right.
– 这一部分数会“爆炸” – 没错
– These ones are not blowing up. – That’s right.
– 而这一部分不会 – 没错
What’s happening at the edges, then?
– 那分界线上会发生什么呢
Is that where things are interesting?

– That’s where things are interesting, right?
– 那儿会发生有趣的事 对吧？
Where you go from blowing up to not blowing up,

is, is dynamically interesting

and just sort of, loosely speaking,

the reason why is that you can’t predict what’s going to happen

if you change C a little bit, right?

So if I have some C on the boundary here,

so it so happens that 1/4 is on the boundary,

if I move that C around by a little bit,

anything can happen, right?

You might have your orbit blow up,

you might have it not blow up.

And so you can’t predict what happens

when you move your C around a little bit

And that’s why it’s interesting.

［Music］
［音乐］
[fading in] And all of these separate disconnected pieces
[淡入]（作填充的朱利亚集）所有得到分散图形的C值（不属于曼德博集合）
and so it turns out

that another way you could define the Mandelbrot set

is by which of these two behaviors you get.

When you draw the filled Julia set for Z²+C,

do you get kind of one piece, one blob?

Or do you get a bunch of disconnected pieces?

So if you get one piece, one blob,

you’re in the Mandelbrot set.

##### 译制信息

https://www.youtube.com/watch?v=NGMRB4O922I