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曼德博集合

The Mandelbrot Set - Numberphile

So, today I want to talk about the Mandelbrot set.
今天我们来谈谈曼德博集合
But I want to—
首先要说明的是
so there’s so many videos and websites and Java applets
有很多视频 网站以及Java程序
and all of these things where you can see the beauty of the Mandelbrot set.
都能带你了解曼德博集合之美
And this really nice fractal picture
看这张漂亮的分形图
and you can zoom in and see all of the interesting things.
若想看看有趣的细节 不妨把它放大
[Music]
[音乐]
What I want to talk about is: What is this object?
我们就来谈谈:什么是曼德博集合?
So, what is this picture a picture of ?
这是一张什么样的图片?
Why do we care about this picture
为什么我们更关注图片本身
other than just its intrinsic, sort of, appeal?
而不仅仅是其固有的某种魅力
And, so just to generally talk about the way that
既然这样 那我就简单说说
maybe mathematicians will look at the Mandelbrot set
数学家们是怎样看待曼德博集合的
[Music]
[音乐]
The first thing we need to understand is
首先要说
that this entire thing is happening
认识曼德博集合 我们要走进
in the world of complex numbers, okay?
复数的世界里
So, if you remember, complex numbers and the complex plane,
还记得复数与复平面吗
the way that we view this is
研究复数
that we have two axes
我们要去建立两条轴
and we plot on this plane numbers of the form,
然后将复数标在复平面上
say a+bi,
例如a+bi
and here these two things are real numbers
其中a b是实数
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.
讲后续内容前 提一下这些基础知识
It’s a convenience in some sense,
有时用这种方法表示复数很方便
but there’s also a lot of useful information in this representation.
不仅如此 该表示法还提供了很多有用的信息
So for example, one thing that’s very natural
比如说 很自然的是
to look at is if I plot some complex number
如果我标出某个复数的话
say a+bi,
例如a+bi
Okay, maybe this is something like you know 3+2i, something like that,
好吧 或许是3+2i之类的
then one sort of natural quantity associated with this thing
自然而然的 相关的一个量
is the distance from this point to the centre point of the plane.
是从该点到复平面原点的距离
And so this distance,
这个距离
which we call the magnitude of the complex number,
被称为复数的大小
it really has some inherent mathematical properties
复数的大小有一些固有的数学性质
that we really care about
是值得去关注的
and so the fact that this is so easy to visualize in the complex plane
在复平面上 复数会变得很形象
you can also visualize like addition of complex numbers and subtraction and so on
将复数的加法 减法等
in this plane in a very geometric way is really helpful.
在复平面上用纯几何形象化有益于处理问题
[Music]
[音乐]
So how do we get to the Mandelbrot set from here?
那我们如何将这种方式和曼德博集合相联系呢?
So here’s sort of just the naive definition.
下面说个简单一些的定义
Let’s take a complex number C
取一个复数C
and let’s associate to this complex number the following function:
把这个复数作为下列函数的参数:
so this is a function which takes as an input some complex number Z
这是一个将复数Z作为变量的函数
and outputs Z squared plus C
输出为Z²+C
So I’m thinking of this complex number
我就将复数C
as being associated to this function,
作为该函数的参数
and we what we’re interested in is
我们感兴趣的是
the behavior of 0 under iteration
0在迭代下的特性
So by iteration of fc(Z)
所以用fc(Z)对0进行迭代
I mean what happens when I take 0
也就是把0
and I plug it into this function,
代入这个函数
and then I keep doing that to the result.
算出的结果也不断代入该函数
So, for example, if we’re looking at f₁(Z)
例如 首先看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?
就是把刚得到的答案代入 对吧
This is 1+1 which is 2.
结果是1+1=2
f₁ of the previous thing
再把前面的数代入f₁
which was 2 is 2²+1 which is 5
也即代入2 可得2²+1=5
f₁(5) is 5²+1 which is 26, and so on.
然后是f₁(5)=5²+1=26 等等
So that’s what I mean about
这就是我说的
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
曼德博集合关注的是
is what happens to the size of these numbers
这些数字的大小会怎样变化
and by size I mean exactly what we were talking about before
这里的大小是指刚刚讲到的
about the distance from the number in the complex plane to this point 0, okay?
复数在复平面对应的点到原点的距离
So it turns out there are two options for a function
对于函数fc(Z)=Z²+C
fc(Z), defined to be Z²+C:
结果有两种
The first option is that the distance from 0
第一种情况是迭代结果与0的距离
of the sequence we get, gets arbitrarily large.
随着不断迭代 会变得越来越大
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?
你想多大它就能多大
So, this is what people mean when they say that the iterates go to infinity, okay?
这就是人们所说的“迭代趋于无穷大”
They mean not necessarily that, okay,
这里说的数并不一定是
they look like real numbers or integers or something like this,
实数或者整数
but that the”size” of the number,in this sense, goes to infinity.
但数的“大小”一定会趋于无穷大
The other thing that can happen is that the distance is bounded.
另一种可能的情况是距离有界
The size is bounded.
即大小是有界的
So, and in fact you can say
事实上 可以说
that it never gets larger than 2.
它永远都不会大于2
So you have this sort of dichotomy
总的来说 你会面临两种可能
where only one of two things can happen
两种情况仅会发生其一
If you give me a complex number C
如果给出一个复数C
and I start iterating zero under that function Z²+C
然后用函数Z²+C对0进行迭代
either the distance of the iterates to 0 in this complex plane
一种情况是 复平面上 迭代结果与原点的距离
gets really large for all of them,
会变得非常大
so you can’t bounce back and forth, right?
而不是时大时小 对吗
It gets really large for all of them.
会变得非常大
Or, it stays close to 0,
另一种情况是 它会呆在0附近
within a distance of 2 from 0.
与0之间的距离不超过2
So, for example, to illustrate these two cases,
例如 为了展示这两种情况
we wrote down already a few iterates, under Z²+1 of 0,
我们已经用Z²+1做了几次迭代
and as you can see their size is growing
可以看到数字不断增长
and in particular we’ve got some things that are further from 0 than 2 is,
而我们已得到距原点距离超过2的结果了
and so this C=1 is case 1.
所以C=1属于情况一
But there’s another possibility so let’s look at, well,
但还有另一种可能 我们来看看
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
虽然这可能会有点误导人
so if we look at, say 0,
如果取0
and we start applying this function
代入这个函数
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
再把-1代入该函数
we have (-1)² which is 1, which is 0.
得到(-1)²-1=1-1=0
Oh, wait, okay, but we know what happens to 0, right?
等等 我们知道代入0时会发生什么
It goes back to -1.
它又会回到-1
So these iterates just alternate between -1 and 0.
所以迭代结果会在-1和0之间交替
And so in particular they never get large, right?
因此它们永远不会变大 对吧
So that’s an example of case 2.
这就是情况二的一个例子
So,the definition of the Mandelbrot set, then,
再来说说曼德博集合的定义
one definition of the Mandelbrot set,
曼德博集合有一个定义
which we usually call M, is the set of C,
我们通常将之为M集合 也就是C的集合
complex numbers C, for which case 2 holds.
是所有情况二的复数C的集合
And I’m kind of all over the place here so let’s be clear,
我有点写得太满了 这样会清晰一点
Case 2
情况二
So,in other words,
换句话说
if I look at the function represented by this complex number,
对于一个由这个复数表示的函数
if I look at Z²+C and I start iterating 0 under that function,
对于Z²+C来说 用这个函数对0进行迭代
everything remains bounded.
结果所得数值在一个范围内变动
– 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.
它不会像第一种情况一样“爆炸”
And this is also
还有这个
in case you’re curious how these pictures are always generated
如果你想知道这些图片是怎么画出来的
So if you wanna figure out to draw a picture
即如果你想弄清楚所画的这个分形图中
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
只要用Z²+C去迭代0
And if it takes a long time to get big
如果它过了很久才变大
then you can give it one color
就不妨给它涂上一种颜色
If it gets big really quickly you can give it a different color
如果它迅速变大 就给它涂上另一种颜色
and that’s how you get these shadings.
这样就能画出这些阴影
[Music]
[音乐]
I’ll point out here
我要指出
that everything that’s in the Mandelbrot set
在曼德博集合中的任何数
has to be within distance 2 of the centre, right?
离原点的距离都最多是2
Because of exactly this case 2 thing that I said
因为之前在情况二中我详细说过
that once your iterate is larger than 2,
只要迭代结果超过了2
you’re out of the picture.
就会在这个图形之外
So the inside of this thing, let’s fill this in here,
所以我把满足情况二的数填入
this is what’s known as the Mandelbrot set
这就是曼德博集合
So let’s look at our examples, right?
来看看我们的例子 好吗
So we had two examples, we had C=1
这有两个例子 一个是C=1的情况
and we had C=-1.
另一个是C=-1的情况
So -1, is right here,
-1在这里
is indeed inside the Mandelbrot set.
确实在曼德博集合中
1 is right here, and it’s outside.
而1在这里 它在外面
let me take the easiest example inside of there
不妨取在这里面的最简单的例子
so we look at 0, right? C=0,
看看0的情况 好吗 在C=0的情况下
And we start iterating,well
开始迭代
what is the function associated to C=0?
把C=0作为参数 函数会是什么
f₀(Z) is Z²,
会是Z²
okay, so, let’s start iterating 0,
好 开始对0进行迭代吧
well, 0²=0
0²=0
So no matter how many times we apply
所以不管迭代多少次
the function we just stay at 0.
函数值一直为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
但如果在这里取一个较小的数
It’s a little hard to compute without taking a real number,
不取实数的话比较难算
so I apologize,
我为此道歉
but if we we take something like 1/8, something like that
如果取1/8 或其他类似的数
if we start iterating so the first iterate is 1/8
开始迭代后 第一次迭代会得到1/8
and then you start adding things under iteration
且随着不断迭代结果会递增
but, but it’s never enough to get you
但永远不会超出
outside of that disc of radius 2.
这个半径为2的圆
– 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?
那儿是不是会发生有趣的事?
– 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,
如果处在“爆炸”和“不爆炸”的分界线上
is, is dynamically interesting
会有一些有趣的动态特性
and just sort of, loosely speaking,
简单来说
the reason why is that you can’t predict what’s going to happen
原因就在于 如果把C改变一点点
if you change C a little bit, right?
你将无从知晓会发生什么 对吧
So if I have some C on the boundary here,
如果在这个边界上取一个C
so it so happens that 1/4 is on the boundary,
碰巧1/4就在边界上
if I move that C around by a little bit,
如果我把C移动一点点
anything can happen, right?
什么都可能发生 对吧
You might have your orbit blow up,
它的迭代结果可能会“爆炸”
you might have it not blow up.
也有可能不会
And so you can’t predict what happens
把C移动一点点时
when you move your C around a little bit
你无从知晓会发生什么
And that’s why it’s interesting.
这就是它有趣之处
[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
这样就得出了
that another way you could define the Mandelbrot set
另一种基于这两个特性的
is by which of these two behaviors you get.
曼德博集合的定义
When you draw the filled Julia set for Z²+C,
对于Z²+C作填充的朱利亚集
do you get kind of one piece, one blob?
你所得的是一整块图形
Or do you get a bunch of disconnected pieces?
还是一堆分散的小片图形?
So if you get one piece, one blob,
如果得到了一整块
you’re in the Mandelbrot set.
那么C就在曼德博集合中

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视频概述

视频中,霍利·克里格博士介绍了曼德博集合的定义,带领我们体会曼德博集合之美。

听录译者

收集自网络

翻译译者

长安小盆友

审核员

审核员 EM

视频来源

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

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