I was later pleased to learn that a computer artist named Lori Gardi, who I had described this technique to several years ago, has since devoted a great deal of her creative effort to generating various high-resolution images using the technique. She named it Buddhabrot which is a name I instantly loved and have adopted. Lori's web site contains some reduced examples of her work along with her writings into the mystical connections she's made between the Mandelbrot set and Buddhism.
The above image shows the overall entire Buddhabrot object. To produce the image only requires some very simple modifications to the traditional mandelbrot rendering technique: Instead of selecting initial points on the real-complex plane one for each pixel, initial points are selected randomly from the image region. Then, each initial point is iterated using the standard mandelbrot function in order to test whether it escapes or not. Only those that do exit are then re-iterated. (The ones that don't escape - I.E. which are believed to be within the Mandelbrot Set - are ignored). During re-iteration, I do not color a pixel according to the number of iterations used, but instead, I increase a count field for each pixel that it lands on before exiting. Every so often, the current array of "hit counts" is output as an image. Eventually, successive images barely differ from each other, ultimately converging on the one above. I'm the most unreligious person you could ever meet, but it's hard not to think of this image as revealing god hiding in the Mandelbrot Set and proving that the Hindus were right all along.
is a 4x magnification of the top of the "head" region of the first
At the very center of this image is the bright area from the forehead
which Lori calls the "Third Eye". It's interesting to note the heart
feature directly below it. It's also interesting to note that the
look of this close-up looks very much like the unmagnified image, which
is a common feature in fractal images.
This image is a close-up
Buddhabrot version of one of the tiny "mini-mandel"
regions floating directly above the head of the main image. You can see
those spots in the high-resolution image linked above. I've not
to link a related underlying mandelbrot image to this one since it is
exactly identical to the first mandelbrot image. You can easily see
differences in the Buddhabrot version however.
This rather unsymetric image is also a Buddhabrot version of one of the tiny isolated mini-mandel islands. This one is floating well off one shoulder of the main figure. It is located at (-1.25275, -0.343) and is only .0025 units in size which is only about one percent the scale of the main figure.
After a long time generating greyscale images I realized that there
is a natural way to use color to
display more information within the Buddhabrot images. Notice
basic Buddhabrot images are generated by choosing a "maximum
threshold just as for Mandelbrot images. One main difference between
two techniques is that Buddhabrot images have distinctly different
depending on the choice of threshold, whereas the effect of different
values for mandelbrot images only changes the amount of black
pixels. I realized that I should be able to generate meaningful color
images by generating three basic images that differ only in the choice
of threshold values, and then combining those images as the red, green,
and blue channels of a single color image. This is exactly the same
that astronomers use when generating "false-color" images of
objects. For example, see the famous Eagle
Nebula images from the Hubble Space Telescope and read the
associated descriptions of color
mages. For my color Buddhabrot images the three
values are analogous to the different frequencies of light which NASA
into their beautiful false-color images. For the image below I used
values of 500, 5000, and 50000, and assigned them to the blue, green,
red channels respectively in order to generate images that most
the NASA nebula images.
Here is a color version of the mini-mandel island above:
After some years I realized how to unify all the Mandel/Julia/Buddhabrot objects and techniques. Click the following link to learn about this interesting extension called the Buddhabrot Hologram.
Still later while exploring different ways to sample and project these sorts of
images, one really surprising thing happened: In one particular rendering projected onto one of the six major planes an image of the logistic map simply popped out! (There are 6 major planes in 4D just like the 3 in 3D.)
Finally, below is a single frame part way through a run generating a low-resolution view of the main figure. Just for fun, I also compiled a set of those frames into an animated GIF file which you can view to see how the image converges over time. Clicking on the image will bring you that animated GIF, but beware since it is over three megabytes which will take a very long time to download completely unless you have a fast internet connection. With a 28.8K modem this will be around 30 minutes. the good news is that you can start watching the animation as it streams across the net to you. Once it's fully downloaded, it will loop quickly through the frames and you can watch the Buddha come swimming out of the void. The complete animation took around half an hour to generate on a P233 Laptop PC.
Alex Boswell found an almost magical way to vastly speed the rendering
of highly zoomed regions. Click here for
Albert Lobo has produced a simply gorgeous music video exploration of the Buddhabrot in 4D. View it on YouTube or download the 22 MB high-res version.
Reimplementations and Ruminations by Others