[MUSIC] Welcome back. Let's take a look now at two of the things I promised to tell you and haven't shown you yet. One of them is fractals, and the other one happens to be strange attractors. I stated that they really are closely tied, intimately tied with chaotic dynamics. To see how, let's return to this picture that I have. Remember this picture from Newton's method where I showed you how to find regions where you prescribe the future? And indeed it turns out that there exists a point that will have that future. What we did is we found first the a small segment of region A that will hit all of region C. But take a look in region C. Now I have a little circle right around that zero in region C. And that's the region where everything goes quite well folks, thank you very much, thank you Mr. Newton. That's where I'm gonna have convergence. That's the good behavior and if we're trying to find wild behavior. So when I find that target region in A, that hit C, I wanna take out a little bit of the middle of A. Why? Because that is the part that's gonna land right in region C where everything goes fine, where you're gonna converge real rapidly to the zero. No wild behavior there so we're gonna take that out. And then what am I gonna do next? Well, in what's left of the wild region of C where can that go? Part of it can go to B, as I showed you how to find. Part of it can go back to C. Just follow what I showed you and you'll find how to do it. Part of it can go to A. Each of those are going to have nice open regions where everything is converging. So I want, in that region, the part of region of C, I wanna take out open parts. And then I wanna take out the open parts in A. Oh, sounding a bit abstract. Sounding a bit abstract. So lets take something simpler and do this and I'll show you where the dynamic crops up. Now what you see is an equilateral triangle. What do we have in this equilateral triangle? It's divided into four smaller equilateral triangles. Much like the story I just told you, now take out that middle. See, what did I tell you in region A? I said take out that middle segment. Here let's take out the middle equilateral triangle. Just throw it out. And all I have left now are three equilateral triangles. I'm going to leave the boundary up there. I'm just taking out the inside of that equilateral triangle. What's next is of each of those three quadrilateral triangles do the same. In each of them I'm going to divide it into four equilateral triangles. In each of them I'm going to take out the middle. Only the middle. And so each of them, three more are left behind. Just like in the story of Newton's method, what am I going to do? I'm going to continue extracting good regions. The middle equilateral triangle. How long am I going to do this? Forever, hey we've got a lot of time, let's do it forever. And as we keep going on, my we are getting a very, very attractive looking figure. It sure is wild isn't it? So, What do we have left? Let me ask you the question, what dimension is it? It's not two dimensional. If it was two dimensional, there would be a little triangle left. And I'm gonna take out the center. So it can't be two dimensional. But it sure looks like a lot of lines, so it sure looks like it's more than one dimension. At least one dimensional or more than that. So what will it be? Mathematicians have devised a way to try to compute what is the dimension of something. And with this method of trying to compute the dimension of an object what do they find? This object has a fractional dimension. It's larger than dimension one, smaller than dimension two. So it's a dimension in between. It's a fractional fractal. And there is where we get our fractals. So the fractals are fractional dimensional objects. And I showed you how to do this with what's called a Sierpinski triangle. You can do it on other objects. Take a square. Divide it up into one, two, three, four, five, six, seven, eight, nine. There we go, nine squares, equal squares. Take out the middle. Each of the remaining squares, divide it into nine, take out the middle. Continue there forever. Or take a cube. And what you do is divide it into a bunch of smaller cubes, and always take out the middle. Continue that forever. So the fractals, you can try all kinds of fractals, they're absolutely easy to draw, and they're fun. The question is We understand what are fractals now. That's going to be an object where I keep taking it away so much from the center, or from some part of it, that what I get is a fractional dimension object. What is a strange attractor? Well, that fractal, that Sierpinski Triangle that I had here, is going to be a strange attractor for a particular dynamic. I'm gonna give you the dynamic in one way which is random, then I'm going to give it to you in another way which is deterministic. The random approach is I'm gonna take a die. I'm gonna have vertex a, b and c. Those three are corner vertices. If I roll a die and I get either one or two, excuse me, choose a point anywhere inside the triangle. You've got the point, you decide the point, you've got the point, now you roll a die, if you get one or two I go half the distance from the point you selected towards vertex on the left corner. If I get a three or four, I go half the distance to the vertex on the right corner. And if I get a five or six, I go halfway to the vertex to the top. Start somewhere, use these rules, roll the die. Got a two. So I'm gonna go halfway to the point to the left. Roll the die. Whoa, now I got a four. So I'm going to go halfway from that point towards the four. And I continue. The question is, where will I end up? Well, turns out that after the first die, you can never get back into that center triangle. If I take the big triangle and divide it up into four, after the first roll, the first move, you can never get into that center triangle. After the second one, you can never get into the center triangles of the outer edges. After the third one, etc. So what you're doing is [COUGH] the dynamic will approach the Sierpinski triangle, this fractile, this strange looking object, this strange attractor. And so strange attractors crop up just about everywhere. Some people have computed the fractal dimension of the stock market and of others in physics. This is a very, very standard tool. So all of these things are tied together. Now we go back to the Newton's method and the same thing is true. At each stage we just instead of looking at what's wild. We take out the middle. The middle where everything is nice and were Newton proved to us, everything's going to converge for certain. We throw that away because it's too nice, we want bad things. So we throw it out and we keep throwing out at each stage and what do we get? We get that the wild, wild behaviour just keeps oscillating back and forth within this thing that has a bunch of holes. A very holey, not religious, but bunch of holes inside there, a holey region. And which is going to be indeed a strange attractor a fractal. And with that, I suggest that you try out some of these drawings in your own. Try a couple of drawings where you know you're going to have a recurrence and an expansion. Try to figure out exactly what can go right? What can go wrong? What will be the strange attractor? With that, I thank you very much.