Determine coordinates for Mandelbrot set zoom. I am writing a computer program to produce a zoom on the Mandelbrot set.  The issue I am having with this is that I don't know how to tell the computer where to zoom.  As of right now I just pick a set of coordinates in the complex plane and zoom into that point.  Only problem is that eventually it will either become completely light or dark because those coordinates are not exactly on the border of the fractal.  Any ideas for how to calculate the "most interesting" coordinates?
 A: One strategy to find some interesting (at least to my own subjective taste) places (which might not make interesting videos) semi-automatically is to spot patterns in the binary expansions of external angles, extrapolating these patterns to a greater length, and trace external rays (rational angles/rays land on the boundary in the limit if I recall correctly) until they are near enough to switch to Newton's method.  Or use the spider algorithm (which I don't yet understand, nor do I know of any arbitrary precision implementations).
Define $f(z, c) = z^2 + c$.  You can find a minibrot island nucleus satisfying $f^p(0,c) = 0$ using Newton's method once you have the period $p$ and a sufficiently nearby initial guess $c_0$ for the iteration:
$c_{n+1} = c_n - \frac{f^p(0,c_n)}{\frac{\partial}{\partial c}f^p(0,c_n)}$
You can estimate if your ray end-point $c_0$ is near enough by seeing if $|f^p(0,c_0)| < |f^q(0, c_0)|$ for all $1 \le q \lt p$.
I wrote a blog post on "navigating by spokes" with one binary expansion pattern extrapolation idea, later posts describe some other patterns.
A: Quick dirty answer: there are two circles, their boundary being zones of chaos. As a starter, you could explore those areas.
Another alternative would be to use a visualiser and a click tracker so you can select the coordinates using a mouse.
