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?