Hello everybody, this unit is on fractals. Fractals are objects that are self-similar at different scales. Think of a tree - it has a trunk that has branches and those branches self have branches coming off of them and those have sub-branches and so on. So that's a fractal. Fractals turn out to be a good way to describe many objects in nature. In this unit we'll explore fractals, both visually and mathematically and I'll survey some of the roles played by fractal geometry in complex systems. Fractals can be intuitively defined as objects with self-similarity at different scales. It's striking how many natural objects there are with this kind of property. Let's look at a simple example. Trees. Trees are fractal and let me explain the notion of self-similarity. Let's take a picture of a tree. Now we'll take part of that picture and crop it out and blow it up. You can see that the structure of this blown up part of the picture is very similar to the structure of the whole picture itself. Let's do that again. Let's take a part of this picture within this red box and let's blow it up. Again, the structure that you see inside this blown-up part of this picture looks very similar to the previous two pictures and notice that this third picture is a very tiny part of this original picture. Now we can do that same thing again. Take a part of this third picture, blow it up - again we have structures that look very tree-like, so no matter how far down you go up to a certain point in these pictures you can keep taking little crops, blowing them up and seeing that they have very similar kinds of structure. That's the crux of the notion of self-similarity at different scales. Now before I go further, let me make a quick note for all of you mathematicians out there. The actual definition of fractal means that the object is perfectly self-similar at all possible scales. So the objects that we're going to talk about in nature are only fractal-like. They're not real fractals from the mathematical sense, but I'm going to use the term "fractal" to describe them anyway. So here's a picture of a special kind of broccoli that has fractal properties. You can see that each of these little broccoli mounds consist of other little mounds that themselves have the same structure and so on. Leaf veins are fractal in the same way that trees are fractal. Galaxy clusters can be fractal. This is actually a cluster of clusters and if I looked at one of those clusters, that itself would be a cluster and so on. So galaxy clusters can be fractal. Plant roots have tree-like fractal properties. Mountain ranges are fractal. If I look at any one of these mountains it also has peaks and valleys the same way that the whole mountain range does. Surprisingly the World Wide Web is a fractal. This is a map of part of the World Wide Web - the connections between webpages and you can see if you blow up at tiny part of it that it has the same kind of spike coming out of a central hub that the whole web picture does. So we'll see later, when we talk about networks, the significance of the fractal properties of networks like the World Wide Web. Those are a small sample of some fractal-like objects in nature. To talk about little bit of history - many mathematicians studied notions related to fractals, such as the notion of self-similarity or fractional dimension and we'll talk about that also a little bit later on and of look like what objects with fractional dimension would be like. So that's going back many centuries in mathematics, but the term "fractal" itself, to describe such objects was coined by Benoit Mandelbrot, a twentieth century mathematician. He took the name from the latin root for fractured. Now Mandelbrot's goal was to develop a mathematical theory of roughness to better describe the natural world. Typical geometry, the mathematics of simple structures describe smooth objects, but the real world actually consists of very rough objects like mountain ranges and galaxy clusters and so on. And Mandelbrot realized that he needed to bring together the work of many different mathematicians in different fields to create a new sub-branch of mathematics, called fractal geometry. Mandelbrot's most famous example is the notion of measuring the length of a coastline. He particularly looked at the coastline of Great Britain, because of it's ruggedness and his question was: Suppose we want to measure it's length - what size ruler should we use? Well, if we look over here we are measuring it with a rather long ruler and we get a certain number of lengths - one, two, three, four, five, six, seven, eight, nine, ten, eleven, twelve. But if we shrink the ruler we actually get a longer coastline. Here we shrink it by half and if you count them up you get one, two, three, four, five, six, seven, eight, nine, ten, eleven, twelve, thirteen, fourteen, fifteen, sixteen, seventeen, eighteen, nineteen, twenty, twenty one, twenty two, twenty three, twenty four, twenty five, twenty six, twenty seven, twenty eight. And twenty eight is more than two times twelve, so that means that the actual length we get here is longer than the length we get here and the reason for that is the smaller rulers can fit a little bit better into the nooks and crannies that we see on the coastline. If we look at an even smaller ruler we cut this one in half. You can see that it fits even better into more nooks and crannies and will give us a longer coastline than we measured here. So what is the real length of the coastline? If we look at an even smaller ruler we would obviously be able to fit into all these other nooks and crannies. The nooks and crannies, as you know, go very far down as we zoom in on the coastline. Let's see a version of this with real coastline pictures. Here I chose the coastline of Ireland. Here's a satellite picture of Ireland. So let's play the same game here that we did with our tree before. We take a small bit of the coastline. We see that this is a fairly rugged coastline, especially over here in the south west part of Ireland. So what I'm going to do is now blow this up and now we can see even more nooks and crannies, so we can imagine sticking a smaller ruler into here and getting a longer measurement for the coastline than if we measured it at this scale. Now I can do the same thing - take a little bit of this and blow it up. I do that on the next page where I blow up this bit. Now I moved from using a satellite picture to a Google maps picture and now I can see even more little tiny nooks and crannies in the coastline. Do that to this little bit over here. Now remember that's not the same as this little bit over here. It's a little tiny bit of that little bit and take that little bit, blow it up. You can't see it as well now, but you can see now even more detail than on here. Even more roughness on a smaller scale. Now we can keep doing this over and over again. Here's this part blown up. We can see more detail and if we blew up a little part of this. Google maps actually doesn't go this far, but you can imagine that now you can get it to a small enough scale where you, a person, could be looking on the beach and seeing this detailed ruggedness and then if I took an even smaller bit of this, I get the level of a crab's point of view. The crab would see even more detail than the person and so on and so on and it keeps going on and on and on. So the question of how long is the coastline of Great Britain or Ireland really depends on the length of the ruler you used to measure it. Now that seems counter to what we might normally think. We think that the length of some object is a well-defined notion - it has a real length. But what is the real length here?