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?
[Math] Determine coordinates for Mandelbrot set zoom.
fractalsgeometry
Related Solutions
Based on the C code given there, Zx is just the real part $\Re z$ and Zy is just the imaginary part $\Im z$.
Remember that the complex iteration formula for the Mandelbrot set
$z=z^2+c$
when expanded to real variables looks like
$x=x^2-y^2+u$
$y=2xy+v$
where $z=x+iy$ and $c=u+iv$.
The variables with a 2 appended are the squares, as can be seen from how they were defined, so for instance $|z|=\sqrt{x^2+y^2}$ is sqrt(Zx2+Zy2) which is used for determining how much the iteration diverges (escapes to infinity), which is used by the coloring functions.
As this graph shows, the behaviour of normalized iteration counts of points near the Mandelbrot set varies widely, indicating that attempts at a formula based on scale factors is doomed to fail.
In any case the ideal number of iterations is infinite. For views with no interior regions visible it is preferable to structure computations such that they can be incremental without a fixed iteration limit, as all pixels will escape eventually. For views with interior regions visible one needs a limit, but this can be set dynamically by considering the behaviour of the pixels - maybe keep doubling the limit until no more pixels have escaped. Interior checking can help speed this up.
How to colour the iterations once you have calculated them is an aesthetic matter that has been addressed by the accepted answer. I do also recommend distance estimation as a way to make filaments uniformly visible.
Best Answer
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.