For some reason fractals have never excited me as much as they do most people. Yeah, there's lots of pretty pictures, but I don't see the math very clearly. (I've had a similar reaction to the notion of 'ethnomathematics' - great idea, but the material is often flimsy.) Ron Eglash's TED talk satisfies me on both accounts - I can see the math!
[Hat tip to another Dan, who blogs at Math for Love, for his pointer to this video and much more!]