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
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"
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
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:
- the 6 major planes,
- sampled points in the m-set vs. out, and
- m-brot, b-brot, and b-gram rendering mode.
Do you recognize this image? I recognized it immediately from james
gleick's book Chaos.
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
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!