Visualizing the Complex 4D Mandelbrot+Julia Set

by Melinda Green

The most famous fractal of all is called the Mandelbrot set. The top right image is an interactive applet that lets you explore this fractal. You can expand any region of interest many times up to the precision of your computer. Clicking the top left image will show you a variety of forms you can find in this object if you know where to look. You can easily make more beautiful images of your own using the applet.

By its nature the standard Mandelbrot set is really just one way to view a 2D slice out of a particular four dimensional object. This 4D object doesn't have a nice name but we can call it "The Complex 4D Mandelbrot+Julia Set". By selecting a different 2D slice in the applet's drop-down list, you can explore some other possible slices through this master object.

Clicking the image on the lower left will take you
to a page
describing
a variation on the standard way this object is viewed called the
Buddhabrot
technique. The important thing to remember here is that** Buddhabrot
images
are not generated from a different fractal formula**. They are
generated
from the standard Mandelbrot formula unchanged; they just *view*
this
familiar fractal object in a new and equally natural way.

Most recently I've realized that a natural extension to the Buddhabrot technique can produce images from the full 4D space. I call images using this extension "Buddhagrams" because their relationship to standard buddhabrot images is something like the relationship between photographs and holograms.

For programmers, I've included the C source code for one implementation of my color allocation algorithm here. All the images here use a type of histogram equalization to to assign color indices to raw image data. Note that the actual colors used is unimportant. Any alternate color scheme can be substituted later and the fine detail will be preserved. The main thing to notice is that even though only 256 colors are being used, the large smooth areas of the image still show even the subtlest gradations of color.