How does mathematics fit into fractal generation for computer graphics? I have to do a research paper on any mathematical concept. The mathematical concept must be complex, so I thought fractals would be a good choice (I was told it was a complex idea).
I have been researching for days on fractals, but I can't figure out what type of math fractals is involved in. I need to research that type of math, understand it, and demonstrate it by finding a solution to a problem I create myself.
I do know about a couple of fractal sets such as the Mandelbrot set, and Julia set. I have seen their equations. How would I fit that into computer graphics?
I researched around, and computer graphics are generated by mid-point displacement or diamond-square algorithm - but they are both simple steps, they don't involve any complex mathematical ideas.
What type of math is involved in fractals? Recursion, complex numbers, functions? Also, how would they relate to computer graphics generation?
 A: The sorts of fractals that are used in computer graphics tend not to be mathematically interesting.  When people use fractals to generate, say, a tree or a mountain, they're not really replicating the behavior of a tree or mountain in a nontrivial way; they're just making something that looks more like a tree or mountain to a person than the last generation of art did.  
So far as I know there are basically two ways of using fractals to actually learn things:


*

*If you're studying the behavior of a nonlinear function under iteration.  This is for instance where the Mandelbrot set comes from.  There are applications to things like number theory here.  Google "complex dynamics" for more along these lines.

*If you're studying a real-world object with rough edges and you want to understand the "amount of roughness."  (The "rough edges" here might not be physical -- applications have included line noise on phones and the behavior of the stock market.)  Google "How long is the coastline of Britain" for more on this.
A: *

*solving complex equations ,  using numerical methods : Newton 

*group theory 

*combinatorics 

*many, many more ...
