The Mandelbrot Set is widely viewed as beautiful and intricate, although I can't give a mathematical definition for those.
The imperfect self-similarities are no accident. Many of the pieces correspond to the behaviors of the critical point $0$ under iteration of $z \to z^2 + c$.To each point in the plane, there is a corresponding Julia set, and the relationship of the point to the Mandelbrot set indicates some of the structure of the Julia set.
