Exercise: Zooming into the Mandelbrot set

Most of the time, as you zoom into an interesting spot in a fractal, the coordinates of the center seem nothing more than a jumbled mixture of random digits. If you type in a regular series of digits, and zoom in with the Zooms control (in the Set Location window), you almost invariably end up looking at a screen filled with all one colour. However, there are a few spots on some fractals where extremely simple coordinates lead you into areas that, while not the most fascinating areas, remain detailed no matter how far in you go.

Type the following locations into to the Real and Imaginary text boxes of the Set Location window for a Mandelbrot set, and use the Zooms control to head down inside:

Location Real Imaginary
Top o' the world and end of the set 0.0 1.0
Way out west -2.0 0.0
Stem of the pear 0.25 0.0
First vertical cusp -0.75 0.0
Second vertical cusp -1.25 0.0

(For the last three locations, you need to set the max. iterations value quite high.)

It is interesting to note that despite the incredible "chaoticness" of the Mandelbrot set, some points are very clearly and concisely defined.