You probably know that the buddhabrot technique produces a different view of the mandelbrot set or other related fractals. To recap: the traditional images color each pixel based on how long it takes to discover the exit status of a coresponding grid point in the complex plane. The b-brot technique displays a map of the popular regions in which randomly chosen complex points dwell during iteration, but only for points known to escape (i.e. not in the m-set). here's a typical rendering:

Another view would do the same but for only initial points known to be in the m-set. This so called anti-buddhbrot view is actually nearly ideentical to what you get if you don't classify points at all because the resulting images would be dominated by the long trajectories of the points in the m-set. here's what you get if you look at the points in the m-set:

Images like this one had been independently discovered by Clifford Pickover long before I thought to try it but he'd missed the crucial idea to filter out the non-exiting points.

So far, all of these images were sampled from and projected onto the Zr, Zi plane. After exploring this for a while, I began exploring a couple of new twists: one was to sample random initial points from the entire 4D space of Zr,Zi,Cr,Ci and projected onto each of the 6 major planes in this space. I called these "buddhagram" images.

Those images are all different and interesting. particularly the last one in which the original mandelbrot image appears:

It seems straightforward to see why this happens. When an initial C point is selected and iterated, the corresponding pixel counter is incremented once for each iteration. if one sample is taken for each pixel, then the image should match the original mandelbrot rendering. One side puzzle for me is why this only works in buddhagram mode and not in the original buddhabrot technique. i.e. this doesn't work when the initial Z values are set to 0 rather than also starting randomly. the resulting renderings show just a single dot at the origin. that seems backwards to me because non zero initial Z implies all the julia sets and not just the special mandelbrot case at Z = 0. so one question for you is how to explain this.

Moving on, I spent some time in 2003 imaging and exploring each of the permutations of
• the 6 major planes,
• sampled points in the m-set vs. out, and
• m-brot, b-brot, and b-gram rendering mode.
The results have been very interesting. This process has produced quite a few nice forms, but one of them rather stunned me. This was the one with points sampled from the Zr,Cr plane, using the original Buddhabrot technique, and which do not exit (i.e. are inside the m-set). here it is:

Do you recognize this image? I recognized it immediately from james gleick's book Chaos. I was rather stunned because Gleick's plot was generated from a simulation of predator/prey populations which seemed to have nothing to do with the m-set. Rather, he called it a plot of "period doublings". More precisely, the image above is a somewhat smeared-out version of his diagram, but as you raise the maximum iterations threshold it approaches that diagram as closely as you like. the above image used a cut-off value of 200.

The Wikipedia page on the Mandelbrot Set contains a section titled Basic Properties which includes a diagram showing the corespondance between the normal Mandelbrot images and the logistic map. I did not understand the math, so my main question was "Why does this particular buddhabrot parameterization produce the well-known bifurcation diagram"? Should this be a mostly obvious result similar to finding the well-known Mandelbrot image among the buddhagrams or does it show us some new relationship? Eventually Piet en Gilberte came forward with an explaination and a beautiful animation to illustrate it. He then added both to a new section of the Wikipedia page called "Relation to the Logistic Map". Thank you, Piet!