Fractals

[Mats]

Sometimes in my spare time, I mess about with fractal pictures. I've put the nicest ones I have found on a free website here:

http://www.fractalarts.webs.com/

I thought if I'm doing it anyway, I may as well share the best stuff with anyone who is interested in some nice fractal pictures.

[NoQ]

Yeah that's a nice thing. I think almost everybody tried writing a simple program to paint the mandelbrot set. Yet, even the Cantor ternary set is still full of mysteries. In fact, fractals are much more than amusement in mathematics, for they serve both as a source of weird counterexamples and as actually working models of many things.

[Mats]

It amazes me how everything from leaves to coastlines show fractal patterns in themselves. It's like the universe is made up of fractals!

[NoQ]

coastlines

Not really a good example. Here we just say that "a typical continuous function is fractal", which is not really essential. Much like stock charts. But

leaves

are the essence, i think. They represent the emergence of beautiful complicated behaviour from simple natural rules.

[stiv]

coastlines

Not really a good example.

It was a pretty good example for Mr. Mandelbrot. How long *is* the coastline of Britain?

[NoQ]

Well, of course they *are* fractal, but this doesn't allow any further investigation. They are "nameless". We wouldn't try to write out the equation of the coastline of Britain, but we can explain the shape of a leaf (:

On the other hand, even the Weierstrass function has troubles with finding out the length (note: he lived in the XIX century) (it became even more obvious when Lebesgue noticed that every function of bounded variation has a derivative almost everywhere, which is also a pretty old result). Hausdorff dimension was invented in 1918.

So i'm currently confused. How do we exactly describe the breakthrough made by Mandelbrot, apart from inventing the Mandelbrot set itself?