Clever fast Mandelbrot - please explain!

[David Hayden]

I wrote a very fast Mandlebrot plotter in the late 80's for the PC. One very effective optimization was to consider a rectangular area. Divide it into 4 sub-rectangles. Now compute the value at the 9 points of intersection (top left, top center, top right, middle left, middle center, middle right, bottom left, bottom center, bottom right).

If all 9 points generate the same value then assume that the entire rectangle is that value. Otherwise recurse on the sub-rectangles. While this optimization isn't always correct, in practice, it provides outstanding results.

Some day, I'd like to port that code to the Prime.

As I recall, I also realized that the Mandlebrot set and the Julia set are the same equation, but using two different planes of a 4-dimensional space. My program lets you plot the equation on different planes, resulting in some really cool effects.