I can't speak intelligently about it or related fractals, but I was watching a program on public tv about the set, and a mathematician said "I'm not going to sit here and tell you that fractals are 'cool'. I AM going to tell you that fractals are USEFUL."; I always liked that quote. So the answer is that it's not just a pretty picture.
But yeah, it, like, explains trees and sunflowers and lightning, maan.
So what is an interesting topic to write about on fractals? Not something that would have to be book length.
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Anonymous2009-11-01 18:40
Fractals are useful for applications, such as a cell phone antenna (I think I saw the same PBS doc about it) but don't really help expand mathematics in any way (ie they haven't helped to answer any questions that existed before they did.) So they're of most interest to an engineer or other such applied math person.
As for a topic for your school paper? How fractals are generated would be a nice little paper. Failing that, how they show in nature & modern technology would also be cool.
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Anonymous2009-11-01 20:16
The Mandelbrot set is important in mathematics. It has applications in systems modeling and chaos theory. Chaos theory has shown up in popular math from time to time like in Jurassic Park, so a lot of people say "cool" without understanding what chaos is. Simply put, a chaotic system is one where a small change in the initial conditions (e.g. whether or not a specific butterfly in Brazil flaps its wings) produces a large change in the final conditions (e.g., whether or not there's a tornado in Texas). Sometimes changing the properties of a system will change its behavior from chaotic to predictable. The Mandelbrot set is a "map" of which parameters are stable and which are chaotic in a certain simple system involving quadratic equations. The points inside the set represent stable systems, and the ones outside represent unstable systems. More complicated systems can often be decomposed or modeled using components which are based on quadratic equations, making the Mandelbrot set more useful than it at first appears.
An example application of chaos is cryptography. In cryptography you often need to generate secret numbers. If you sample the output of a chaotic system, anyone trying to guess your number would need to know every detail of its state (such as whether your butterflies were flapping their wings). If you sample a non-chaotic system, then it is much easier for someone to guess your secret number because it's more predictable. An example chaotic system is a set of three oscillators connected in a loop, an example nonchaotic system a clock... which many people use as seed values for generating secret numbers, making them vulnerable to attack.
Fractal geometry in general has even broader applications across practically every scientific field.
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Anonymous2009-11-02 11:07
Oh my God. I've got some fucking Jaffa cakes in my coat pocket!