This is a Julia set, a fractal. You can show the self-similarity of this fractal by using the drag-and-drop tecnique to define the rectangle you would like to see enlarged.

Again, you can graph either the basin which defines the set or the contour to see the pattern of points which are "nearer" to the set.

The mathematics here is very similar to that of the Mandelbrot set in that the same fixed point c is used for the entire computation.