These fractals are derived from methods for finding numerical solutions to equations. The Convergent Fractals page in the Technical section has information on the mathematical details. The fractals show how quickly each initial estimate converges to a solution.
These fractals are very nice, but the underlying equations are not complex enough to give a good variety of structures at higher magnifications. That means the fractals are truly selfsimilar, and look exactly identical at all levels of magnification, so there is nothing more to see by zooming in.
By choosing different functions to solve, we can create different fractal images. One of the earliest fractals made many years ago was based on finding values of z that have z^{3}1=0, which are the cube roots of 1. In the complex number plane, there are three values that solve this equation, and depending on which initial starting point is chosen, Newton's method will converge to one of these three roots.
I have made a few images applying Newton's method to the sine function, and also to finding the 10th roots of 1.
Halley's method is a more complicated and more rapidly converging technique that is similar to Newton's method. It works better than Newton's method for its intended purpose, which is to find solutions to equations, but its more rapid convergence means the fractals it generates are actually less interesting.
The Secant Method makes more interesting fractals, but is not as useful in numerical analysis.
Newton's method is the method I chose to use in my highprecision division function. This is a standard approach, as it converges more rapidly than the simpler approaches and doesn't require much more calculation time.
Newton's method applied to z^{3}1=0. This is a classic image, one of the first fractals ever made back in the early days of fractal exploration. 

Zooming in to the center, we see the theme that is repeated infinitely throughout this fractal. 

Zooming in again. Can you tell how deep? This is all you will ever see in this thing, no matter where you look or how much you magnify. 

Each point in the plane converges to one of the three solutions to z^{3}1=0. Here I have made the classic redgreenblue rendering showing which root each point goes to. Getting my program to do this was a hassle, but now I have the ability to create very intricate colorings with combinations of different palettes. Blue represents starting values that converge to z=(1,0). Red and
green represent starting values that converge to the complex roots of 1,
which are What's remarkable about this is how complicated the boundary between the different domains is, and how you can start close to the domain of one root but end up converging to a different one. 
Halley's method for x^{3}1=0. Note that the more efficient rootfinding method gives a less interesting fractal. 

As with the Newton fractal, this is a Julia set type of rendering and will look exactly the same no matter where or how much you zoom in. 