"Chaos theory" seems to be one of those misnomers that smell much more of Greek paganism than rational physics. It is safe to say that "chaos theory" was the last thing that Johannes Kepler needed in order to simplify astronomy for high-school students.
Chaos theory (in my opinion) is more appropriately called a theory of equilibrium. The "order in chaos" comes out of some interference pattern that bring an impressive amount of determinism to seemingly disorderly systems. Fractals stand for series of re-iterated operations that approach the equilibria of chaos theory.
Perhaps the universe still does, in a sense, have a mind of its own - but that would not make it any less rational. If anything, having a mind of its own would probably make it morerational.
If you would like to see physics that possibly smells a lot more like Kepler, Newton, and Occam, I recommend the physics site that I have been developing - along with one other site that gave me my inspiration. I especially recommend the papers where I cover fractals.
I say all these things as a theoretician who really believes he has answers - but must bypass standard explanations.
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But to a mathematician, it is a neat, neat subject area. Why are fractals important? Fractals help us study and understand important scientific concepts, such as the way bacteria grow, patterns in freezing water (snowflakes) and brain waves, for example. Their formulas have made possible many scientific breakthroughs.
Nobody. Fractals are not owned by anyone!
Complex mathmatic equations.
Self-similarity.
Fractals are situations where the geometry seems best approximated by an infinitely "branching" sequence - used, for example, in modeling trees. For work on fractals that I have done as a theoretician, I recommend the included links. I just happen to have an original answer, and I want to make it known.