Here's another way to visualize this 4D object. The original
Mandelbrot rendering method samples 4D points from a 2D slice and
renders a map of the ultimate fate of the iterated samples. The
points determined to be inside the M-Set are typically rendered black.
The standard Mandelbrot images are from the Z==0 plane whereas
Julia-Set images are taken from other slices parallel to the Z==0
plane. A sequence of images can then be made from stepping through
various distances from the Z==0 plane and the collection of all the
black regions can be though of as one solid object. Unfortunately that
object is still 4D so we'll still need to choose a 3D projection in
order to visualize that solid. The simplest projection from 4D to 3D
is just to ignore one of the dimensions. What this means is that
instead of always initializing both Z values to 0, we'll just
initialize one of the initial Z components to 0 and make a sequence of
Julia Set images where the other component ranges from say -3 to 3.
There are therefore 2 such natural 3D projections. One where the Z-real
component is held at 0 and Z-imaginary ranges, and one where
Z-imaginary is always initialized to 0 and Z-real varies.
Ok so that will give us a stack of images with a black 3D form in
the middle, but how to render that 3D object? Well, the best way would
be to use a 3D lithography machine to 'print' a real 3D object. That's
still rather expensive to do so let's try some other approaches. A
simple solution is to create an image from each slice and then make an
animated movie that flips through those images. Here are those two
| Slices at
Z-real = 0
||Slices at Z-imaginary = 0|
An algorithm called "Marching Cubes" can take a 3D data array such as this and produce a 2D isosurface for any given data threshold. Isosurfaces are the 3D equivalent of the familiar 2D 'isobar' weather maps showing lines of identical temperature or pressure. In this 3D case all points on a 2D isosurface represent points with identical exit values. If the black points in the Mandelbrot Set are indicated by 0 values then a reasonably faithful isosurface of the M-Set can be produced using a threshold value of 0.5. Here are renderings of those two isosurfaces produced from the above data slices. You can click on either image to pop up a stereo viewing applet to see these views in 3D. You can also click the following links for the 3D VRML models from which these screen shots were snapped of the Z-real==0 and Z-imaginary==0 isosurfaces. Note that each stereo image is less than half a megabyte but each VRML file is over 2 megabytes plus you'll need a VRML browser plug-in to see and manipulate them.
with Z-real initialized to 0
Click to view in 3D
with Z-imaginary initialized to 0
Click to view in 3D
Not a surface, but another way to view the inside vs. outside nature
of the full 4D object is to compute a lot of random initial 4D points
and plot them as a desnity field. The left above image shows one frame
from a movie showing a rotation of a projection of this field in 3D.
Click the image to see the animation. Each dark dot in this space
represents an initial point determined to be inside the 4D m-set. This
makes it a true 4D analog of the 2D black & white Mandelbrot images
(on right) where the points in the m-set are colored black.
Marching Cubes implementation courtesy of Marcus
3D Applet courtesy of Andreas Petersik