Discuss.
Discuss.
I watched a one hour documentary about fractals on PBS the other day. I would say some of the technologies developed because of fractals make them very important. There's nothing theoretically groundbreaking about them though.
Dunno what you're looking for here, but the concept of fractal dimension is very important in many fields of mathematics. For instance in statistical mechanics, the result that SLE_k paths have fractal dimension 1+k/8 almost surely allows one to determine corresponding quantities for the discrete processes that are known to scale to them.
They look pretty?
From what I've seen, they're an interesting concept. But, I'm not a mathematician. I don't even know what mathematicians do to make money, much less how they'd be useful to them.
Well, according to The Hitchhiker's Guide to the Galaxy there are a number of important things about fractals that we should all be aware of.
As far as practical applications... I'm not sure. One thing that's nice about fractals is that they maximize the surface area of an object for a given volume. So there are a lot of objects that seem fractal-like -- airports, for example, need a lot of surface area for planes to go in and out, but don't want to take up too much space.
If you're asking about the details of the mathematics of (say) the Mandelbrot set -- I'm not a mathematician, but I think people are just interested in knowing what kinds of properties are possible for different mathematical systems.
And it is extremely difficult to obtain funding for mathematics research :)
There is no exact definition of fractal but by most definitions, Brownian motion in dimensions 2 and higher is a fractal set (it has self similarity and the image of any finite interval has infinite length and zero area). Brownian motion has tons of applications in math, science and finance.
Brownian motion is a limit of random walks. Fractals often occur as limits of discrete things; other examples are limits of percolation, limits of polymers or limits of processes that try to maximize length vs area (similar to the airports by previous poster). Here's a cool example of a fractal obtained using a natural construction:
http://en.wikipedia.org/wiki/Diffusion-limited_aggregation
(and for which very little is known).
More common fractals like the Mandelbrot set and the Sierpinski carpet are perhaps only useful to mathematicians as examples that display unusual behavior.
Fraggles are very very important.