Now let's look at some actual applications

of fractal dimension in the real world.

I'm sure you've always wanted to know

the dimension of cauliflower.

Fortunately, it's been calculated

at approximately 2.8.

That is, if you look at a cross-section 
of cauliflower,

this flat thing,

you find that it's actually a little more than 2 dimensional.

It's between 2 and 3 dimensional,

due to the dense fractal self-similarity

of the cauliflower branches.

You may have noticed that

the logistic map bifurcation diagram

has a tree like structure

which indeed is fractal.

So for instance, if we blow up a
little part of it here.

And some parts of it are quite 
self similar.

We blow up this part here, we see that

this looks very much like the whole thing.

And people have calculated the 
fractal dimension

to be about .5.

It has so many holes in it

that it's not even quite 1 dimensional.

The fractal dimension of coastlines

has been calculated.

Notice that the west coast of Great Britain

has a higher dimension than

the smoother coast of Australia.

or the even smoother coast of South Africa.

And all of these are a little bit 
more than 1 dimensional.

If you look at them as going along a curve,

sort of like the Koch curve.

People have looked at fractal dimension

of more abstract kinds of phenomena,

like stock prices.

This is from a paper that was published in 2000

looking at the Oslo Stock Exchange

and looking at an index which listed

a hundred day daily price records,

a hundred week weekly price records,

and a hundred month monthly price records.

And you can see that all of them

have very similar kinds of ups and downs,

even though they're at very different 
time scales.

The person who wrote this paper asked the question,

"Are stock prices following a random walk?"

And the project here was to compare

the fractal dimension of these curves

with the fractal dimension of a random walk

to see if the dimensions were the same.

If you look at random walks,

they also have a lot of detailed ups and downs

and self similar structure.

But after some complicated mathematics

this paper was able to answer "No."

The fractal dimensions are not the same.

Therefore, it's not likely the stock prices

are following a random walk.

Of course, in these real world examples there's no exact

proportion of size reduction or clear cut number of copies.

It's very different from the exact mathematical fractals

like the Koch curve and the Sierpinski triangle.

So there's a lot of caveats in actually applying

fractal analysis to time series.

A group of scientists looked at fractal dimension

as applied to Jackson Pollock's drip paintings.

They looked at the fractal dimension in these paintings

and found that if you plot the year of the work

versus their measurement of its fractal dimension

you get this kind of increase,

and this is claimed to be the evolution 
of complexity

as measured by fractal dimension 
of Pollock's paintings over time.

Using measures like fractal dimension

for quantifying aspects of art has

been a controversial subject for a long time.

Later on we'll talk to John Rundell,

a geophysicist who's been interested in using ideas

from fractal geometry for a long time

in the natural sciences.