The Image Looks So Weird At First. But Then She Goes Deeper, And Suddenly It’s Beautiful.

Eric March Curated by

Disclaimer #1: I am a big ol' geek. Disclaimer #2: This is, like, the geekiest thing ever. It's all about math and numbers and equations and stuff. But the end result of all those numbers and equations and stuff? Kind of breathtaking.

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Narrator: 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 there's really nice fractal picture, and you can zoom in and see all the interesting things. What I want to talk about is what is this object? So what is the 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 the mathematicians would look at the Mandelbrot set.

The first thing we need to understand is that this entire thing is happening in the world of complex numbers, OK? So if you remember, complex numbers or the complex plane, the way that we view this is that we have two axis and we plot on this plane numbers of the form say, a plus bi. And here, these two things are real numbers. And "i" is a symbol that means that i squared is equal to minus one, OK? 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 plus bi. OK, maybe this is something like, you know, three plus two i, something like that. Then, one sort of natural quantity associated with this thing is the distance from this point to the center 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, and 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.

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 what we're interested in is the behavior of zero under iteration. So by iteration of fc, I mean, what happens when I take zero and I plug it into this function? And then, I keep doing that to the results. So for example, if we're looking at f1 of z, well, f1 of zero is equal to zero plus one, which is one, f1 of one, so now I apply it to the answer that I got, right? This is one plus one, which is two. F1 of the previous thing, which was two, is two squared plus one, which is five. F1 of five is five squared plus one, which is 26, and so on. So that's what I mean about the behavior of zero under iteration for a particular value of c. 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, zero, OK? So it turns out there's two options for a function fc of z, defined to be z squared plus c. The first option is that the distance from zero of the sequence we get gets arbitrarily large.

Male: That means it blows up.

Narrator: Means it blows up. It gets as large as you want it to be, OK? So this is what people mean when they say that the iterates go to infinity, OK? They mean not necessarily that, OK, 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 two. 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, z squared plus c, either the distance of the iterates to zero 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 zero, within two of zero.

So for example, to illustrate these two cases, we wrote down already a few iterates under z squared plus one of zero. And as you can see, their size is growing. And in particular, we've got some things that are further from zero than two is, and so this c equals one is case one. But there's another possibility. So let's look at... Well, a good contrast maybe would be z squared minus one. Although, this might be a little misleading. So if we look at, say zero, and we start applying this function. Well, f minus one of zero, that's zero minus one, which is minus one. If we plug in minus one to that function, we have minus one squared, which is one, which is zero. Oh, wait, OK, but we know what happens to zero, right? It goes back to minus one. So these iterates just alternate between minus one and zero. And so in particular, they never large, right? So that's an example of case two. 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 number, c for which case two holds. And I'm kind of all over the place here, so let's be clear. Case two. So in other words, if I look at the function represented by this complex number, if I look at z squared plus c, and I start iterating zero under that function, everything remains bounded.

Male: It's the guys that don't blow up rather than the ones that do.

Narrator: That's right. 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 want to figure out, to draw a picture, whether c is in the Mandelbrot Set or not, well, you just start iterating zero under z squared plus 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.

I'll point out here that everything that's in the Mandelbrot set has to be within distance two of the center, right? Because of exactly this case two thing that I said, that once you're iterate is larger than two, 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 equals one, and we had c equals minus one. So minus one is right here, is indeed inside the Mandelbrot Set. One is right here, and it's outside. Let me take the easiest example inside of there. So if we look at zero, right, c equals zero, and we start iterating, well, what is the function associated to c equals zero? F0 of z is z squared. OK, so let's start iterating zero. Well, zero squared is zero. So no matter how many times we apply the function, we just stay at zero.

Male: So you're in the club.

Narrator: So you're in the club. That's right. But if we take like, say, some small number here. It's a little hard to compute without taking a real number, so I apologize, but if we take something like, you know, one over eight, something like that. If we start iterating, so the first iterate is 1/8. And then you start adding things under iteration, but it's never enough to get you outside of that disk of radius too.

Male: These guys are blowing up.

Narrator: That's right.

Male: These ones are not blowing up.

Narrator: That's right.

Male: What's happening at the edges then? Is that where things are interesting?

Narrator: That's where things are interesting, right? Where you go from blowing up to not blowing up is dynamically interesting. And just sort of like 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.

All of these separate disconnected pieces. 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 z squared plus 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.

There may be small errors in this transcript.

This complex mathematical principle was broken down by Dr. Holly Krieger of MIT for Numberphile. Please do give them a look on Facebook, Twitter, and YouTube for more mathy goodness.


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