by Melinda Green
The images on this page were all generated using a technique I developed in 1993 to render the Mandelbrot set. It's important to realize that it is not a different fractal from the Mandelbrot set, but simply a different way of displaying it. Clicking on some of the images will take you to a normal rendering of the exact same area, but using the traditional Mandelbrot technique. Note that even though the images resemble Hindu art, they were actually generated completely automatically, without any sort of human artistic intervention. When I first tried using the new technique, I had no idea what the images might look like and was completely surprised by the results.
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 or larger as needed. Then, each initial point is iterated using the standard mandelbrot function in order to first test whether it escapes from the region near the origin or not. Only those that do escape are then re-iterated in a second, pass. (The ones that don't escape - I.E. which are believed to be within the Mandelbrot Set - are ignored). During re-iteration, I increment a counter for each pixel that it lands on before eventually exiting. Every so often, the current array of "hit counts" is output as a grayscale 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 not hiding in some tiny corner, but a single image hiding in plain sight at full-size, suggesting that the Hindus were the ones who got it right.
This is a 4x magnification of the top of the "head" region of the first image. At the very center of this image is the bright forehead region which Lori calls the "Third Eye". It's also interesting to note that the overall look of this close-up looks very much like the entire unmagnified image which is common in fractal images.
This image is an extreme close-up of the third eye area using yet another 4x magnification. It was very tricky to generate this image because it's not possible to zoom arbitrarily deeply into the Buddhabrot set as it is with the mandelbrot set. The technique used to create this image was somewhat like clipping out a section from a much larger image. It's interesting to note that none of the features in this image are associated with features in that portion of the underlying mandelbrot set. The entire image area is completely contained within the main cartoid of the mandelbrot set. That's why there is not a mandelbrot image of this region linked to this image: It would be completely black. This image took an entire long weekend to generate. For those familiar with the mandelbrot technique, the size of the image region is 0.125 units, and the coordinates of the image center is (-1.15, 0.0).
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 below. The underlying mandelbrot image for this region is almost exactly identical to the first one. You can easily see substantial 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 the right 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 that basic Buddhabrot images are generated by choosing a "maximum iterations" threshold just as for Mandelbrot images. One main difference between the two techniques is that Buddhabrot images have distinctly different appearances depending on the choice of threshold, whereas the effect of different threshold values for mandelbrot images only changes the amount of black (unresolved) pixels. I realized that I should be able to generate meaningful color Buddhabrot 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 technique that astronomers use when generating "false-color" images of astronomical objects. For example, see the famous
Eagle Nebula images from the Hubble Space Telescope and read the associated descriptions of
color astronomomical mages.
For my color Buddhabrot images the three different threshold values are analogous to the different frequencies of light which NASA combined into their beautiful false-color images.
For this image I used threshold values of 500, 5000, and 50000, and assigned them to the blue, green, and red channels respectively because this mapping generates images that most resemble the nebula images. Some editor of the Wikipedia page labeled it "Nebulabrot", though I don't really like that name. I don't have any other name for it so I guess it will stick.
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 ofthe logistic map simply popped out! (There are 6 major planes in 4D just like the 3 in 3D.) This had me puzzled for years until 2009 when Taneli Hautaniemi also found it and contacted me. In 2010, Piet en Gilberte then stepped in and made a beautiful animation showing the relationship and added it along with explaination to the
Buddhabrot page on Wikipedia. Thanks, Piet!
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 an example.
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.
Benedikt Bitterli later produced a beautiful 4K video with clever speed-ups and denoising. He provides several video frames there suitable for wallpaper. I particularly liked the image to the right which he generously rendered for me in 16K which I had printed and framed. You can get that rendering here.
Reimplementations and Ruminations by Others
