In the previous videos, I have shown you how complexity arises from synchronicity by doing iteration of equations, and now, we would like to think about a new class of things, fractals, where complexity arises not because of iteration on equations, but iteration of shapes. What do I mean? First, let me ask you a question which seems to be unrelated, which is how many dimensions does a curve have? Here, I'm drawing a curve for example, and for you to see easily, I draw it pretty thick, but imagine it's infinitely thin, is just a curve of one dimension, okay? One dimensional object, and I'm asking you, how many dimensions does a curve have? Here for this curve, very obviously, it is only a one-dimensional, it's very clear, but I'm asking you that are other curves having one-dimension, does there exist a curves that having more than one dimensions but parameterized by one parameter? Are there such curves? Yes, they are. At least they use some of the definitions of dimensions that you can imagine. Let's take an example. For example, this curve, okay, this is one-dimensional. This is one-dimensional, but if you do your iteration from one part to self-similar small parts, doing the iteration, many, many, many, infinitely many times as a limit, it is still a curve, but this curve no longer having one dimension because it is subbase feeling in the sense that every point is base, eventually will be occupied, will be passed by by this curve in this infinite limit. So in some sense, this curve becomes two dimensions. One dimension emerges from a one-dimensional parameterization of this curve. This is amazing and this inspires us to think more about dimensions. This curve is known as the Hilbert's curve. It is not so difficult but to understand clearer about the question of dimension, let's think about some, a little bit easier objects, which is Koch snowflake. What is Koch snowflake? Let's do the following thing to generate a shape like this. That first I have a line and I rescale this line into 1/3 of the land, and I put four components into it, and I further rescale the hosting into a third, in length dimensions, and I put four of them in here and doing this iteration over and over and over again. So I got Koch snowflake. Do you see something interesting then [inaudible] this line? Do you see this line to be a bit thicker than this line? Although, I'm plotting in the same line style, and I will make sense of appears to be thicker by counting the dimension of this Koch snowflake is somewhere between one dimension and two-dimensions. Also how long is this line? They can ask, how long is this line? You will get a surprising result. Now let's study the Koch snowflake in more details. We've noticed a few features of this Koch snowflake. The first feature is self similarity. Imagine that I cut this small part of the snowflake. I mean, if I do infinite number of iterations, then I cut a small part, it is similar to the whole part. It is similar to the whole part, and if I can get from the small part to the whole part, how do I do? I just scale it, put three times as large as the original cutting the part, then it will be the same as the original part. So this self similarity is a key feature of fractals. For this Koch snowflake, it is simple, exact self similarity, and for more complicated objects, it may be statistical self similarity. I will come to that points later. But here for now, first of all, you'll notice the self similarity. Second, I'd like to ask you the question, how long is a Koch snowflake after infinite number of iterations? Very clearly, this Koch snowflake is bounded, is a image that bounded within a finite region, but is it finite length or infinite length? We can see it very clearly that it is infinite length. Why? Because we start from a line which is finite length, let's say we start from the last one, and then after the one iteration, how long the line is? Four over three. After second iteration, we get this factor to the square, third aggregation to the Q, and eventually for U infinite number of iterations, we are putting U infinite power of this number, which is greater than Y. As a result, we have a line which is U infinitely long bounded in this finite region. Now, let us return to the question about dimensions. How many dimensions do we have for this Koch snowflake? How many dimensions? We actually have to define a dimension which can work for this very complicated object, the fractal object. With that you define fractal dimensions, and there are many definitions. Here I'm using the host of dimension as an example that, first of all, this is complicated. Let's first think about a very simple object. Think about a square. How do we know, what is the dimension of the area of the square? This is how many dimensional object? Simple question. We know it is a two-dimension, but how do we get two-dimension? We just know it. It's just the ability in your mind. But we do it here a little bit formal way that we subdivide this square into self-similar pieces. We have many different ways to subdivide, but let's take a simple way, and if you would take other ways that's consistent. Let's take a simple way to subdivide and take one of the self similar pieces out, and this one self similar pieces, this one, how should I scale this object to match the original object? How do I scale that? I need to scale that the length is multiplied by 2, then I have this object and this is one thing, scaling. This is scaling. How many self-similar pieces do we have? We have four self-similar pieces, so the scaling factor and the number of self-similar pieces. There is a relation, and this relation can indicate dimension. How do we get two from this two numbers? Four and two. There may be many ways. To think clearer, maybe we think about a cube. We have a better idea that if we subdivide a cube, then, how many pieces do we have? We have eight. Again, the scaling, the last scaling between one self similar piece and the whole thing is two. How do we get three from this eight and two? Now, I think the question is more clear that we do the logarithm, no matter what base and maybe base two or base whatever. Logarithm of this number eight, which is the number of self-similar pieces that we have, divided by the logarithm of how much scaling that we are doing, which is two, and we get the three, which is a cube having three dimensions. This is the idea of dimension that we would like to generalize to this case, inspired by our previous arguments, we would like to define the dimension of fractals as the logarithm of the number of pieces, divided by the logarithm of the factor of scaling. How large we scale, and now it is straightforward to calculate the fractal dimension of Koch snowflake. That dimension is, first of all, how many self similar pieces are there? There are four. This is a logarithm of four divided by, what is the scaling? The scaling from here, I scale this one to here, which is a factor of three, a factor of three, etc., The scaling factor is three. This is the fractal dimension of the Koch snowflake, and clearly it is a dimension somewhere between one and two. That means it is richer, it occupies more dimension than a simple curve, but it is less than two-dimension. In the same way, I hope you can try to calculate that here [inaudible] curve has fractal dimension two, and that is why this curve is a base filling because it is indeed in the sense of fractal dimension, two-dimensional object.