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?