published in: IRIS Universe (summer, 1988)
Creatures of the Complex Plane
The images illustrated here represent a zoology of forms which are the products of
specially designed functions.
These functions may be thought of as stylized sculptor's chisels which are used to
carve into the dense fabric of the
complex plane. The mathematical process known as "iteration in the complex plane"
has recently become a popular topic
due to some striking discoveries. The most celebrated is the Mandelbrot set, a
geometrical object which has an infinity
of detail and a highly organic form. Dr. Benoit Mandelbrot, the founder of
fractal geometry, likened the iterations of
his complex functions to the act of sculpting. The more times the function
is iterated the more detail can be carved
out of the plane. These images are some of my own interpretations of the poetic
qualities latent in the Mandelbrot set.
While the qualities are essentially mathematical attributes, brought about
by discrete operations, they are treated
here as metaphorical. Like many other fractal artists who carve the complex
plane, I am interested in the symbolic implications of this new world of
visuals, a world which was not possible seven years ago. This, in addition
to an inclination toward the surrealist movement in art, and a dedication to
natural form and color, has contributed to the motivation behind these pictures.
Aside from the sculptural aspect of iteration, fractal functions may be thought of
as genetic codes determining the anatomies of the shapes. While exploring the
inner-workings of a function and finding interesting alterations one may get a
sense that he or she is unravelling the DNA helix. To me it is an artform of
abstract biogenetics, and the IRIS serves as an elaborate petri dish. Fractalizer
Peter Oppenheimer of the New York Institute of Technology and others promote such
analogies of fractal art to biological creation.1. For the artist working with
fractals such as the Mandelbrot set, a new method of crafting visuals becomes apparent:
the organization and beauty of the forms are not entirely creations of the artist.
With practically no visual manipulation aside from the actual techniques of display,
fractals seem to have a life of their own. But a computer artist may wish to inject
a personal style into the forms to coincide with the already stunning imagery that
exists by default. This may be done by "tweaking the parameters", either with
clearly defined mathematical transformations or with more subjective, "amathematical"
These deformations of the Mandelbrot set are explained
from the standpoint of a visual thinker concerned with the manipulation of
pre-existing imagery. To begin, as stated, the Mandelbrot function is the
mathematical tool for carving out this well-known fractal form. One may also
think of any alteration of this function as a tool for generating other fractals.
This has been done to an extent by mathematicians and artists interested in
illustrating complex dynamics or exploring new fractal aesthetics. Thomas Papathomas
and Bela Julesz of AT&T Bell Labs have animated deformations of the set as a part
of their research in human visual perception of motion and depth.2.
approach is to a great extent amathematical: the method of generating the images
relies more on intuition and a visual sense than on any clear-cut mathematical
concept. Still, a description of the logical starting point for this technique may
help in describing how the images came about.
The Mandelbrot Set is defined as the
set of all locations C in the complex plane in which Z remains finite during the
Zn+1 = Zn2 + C
where Z and C are complex numbers, and Z0 = 0. C serves as a constant which represents
a location in the complex plane being tested. In short, each C in a given window of
the plane is used in the function, which is iterated a given number of times.
The final state of Z after n iterations determines whether that location C is in
the set or not. A more thorough description of the generation of the set may be
found in Dewdney (1987, 1985), Peitgen and Richter (1986), and Mandelbrot (1983),
among many others. Techniques for colorizing images of the set may be based on
information such as the number of iterations it took Z to exceed a certain magnitude
for each location, or the actual value of Z after n iterations. The former is a
standard way of displaying the set but both are used in the present treatment,
including other methods based on the nature of the series of Z's created by the
In my program, the key parameters used as genetic "tuning knobs" of
the Mandelbrot set are actually real number variables in an expression of
the function with its real and imaginary parts separated:
Zrn+1 = Zin2 - Zrn2 + Cr
Zin+1 = 2 Zrn Zin + Ci
The first of the Mandelbrot set's "freaky others" I discovered was the result of
changing the 2 in the second line to -2. This seemingly small change transforms
the set into a creature having a distinct 3-fold symmetry and exhibiting
complexity only at its extremities. One of the first discoveries made:
it is possible to generate shapes in which the fractal parts (the complex,
fuzzy areas) are shifted to occupy distinct localized areas or configurations.
More varieties of manipulation are made possible by attaching tiny tuning
knobs to the eight variables in the example. For instance, the Zr could be
expanded to Zr+t, or Zr/t, etc. with t being any small fraction entered
interactively. I'll leave the description open-ended at this point since
any further variations one might try would be subject to a particular programming
with these genetic tuning knobs, you might soon get a sense of the Mandelbrot Set
as a piece of mathematical rubber, but a unique kind of rubber subject to the
properties of the complex plane. To me, the complex plane is a rich ground for
poetic exploration. The mathematical explanations behind most of the poetry
came only as an afterthought. And for this reason, the math was revealed in a
more beautiful and metaphorical way than it could have been otherwise. This is
the way some people like to learn mathematics.
Two and a half dimensions
The third dimension is the IRIS workstation's specialty. It is designed to be
proficient at performing transformations in 3D, and fast. While superworkstations
like the IRIS are bringing the 3 dimensions of Euclid down to a hardware level,
non-euclidean geometries such as fractals are giving rise to new methods of
image-making that don't necessarily follow the rules of standard geometry.
Fractal objects can have dimensions which lie between whole numbers.
These images suggest illusionism and depth, but they do not involve any use of
IRIS 3D geometry, as do the beautiful landscapes by Dietmar Saupe. (see last
issue of the IRIS Universe). This illusionism is suggested by utilizing the
smooth gradations created by mapping these functions. The gradations can be
interpreted as topographies of three-dimensional surfaces, as D. Saupe has
done. It is provocative to compare the Z in these mappings (especially as
existing inside the set) to the Z-axis of three-dimensional space. But in these
examples, the analogy is expressed as pure color. I prefer to keep this analogy
loose, for the same reason that painters of the Modernist tradition often provoke
a tension between two-dimensionality and illusion.
In actual practice these
pictures are generated on a two-dimensional cartesian plane, but in
appearance, and also in theory, the dimensions are quite fuzzy. You could
say that these images are examples of two-and-a-half dimensional computer
Although no use of IRIS 3D functions were used, the whole technique would not
have been made possible without the sheer display speed and interactivity the
IRIS offers. This kind of technique requires constant visual evaluation of
parameter inputs. Given control over time-consuming factors like picture
resolution and the number of function iterations, a workable image can be
generated in a matter of seconds. After literally hundreds of trials and
changes involving tweaking parameters, running new colormaps, and adjusting
pans and zooms, things like resolution and iterations can then be jacked-up
for the final running of the image. It is then ready for the camera (a 35mm
camera on tripod, with low ASA print film, and shutter speed set at about
What this discussion may illuminate for those already familiar
with fractal art is a divergent way of approaching the Mandelbrot set.
Most other treatments involve magnifying very remote areas of its boundary
and then applying wild colorations to the image. This approach does not
interpret the Mandelbrot set literally. It begins with the set as a biological
creature, with a genetic code, a physiology, even a personality, to be
manipulated (with mathematical intrigue but not rigor), and then displayed
in its whole using naturalistic colors.
The Advanced Graphics Research Lab at Syracuse University
This work was produced on an IRIS 3020 at the Advanced Graphics Research
Lab at SU, an all-campus research facility with an emphasis on image processing
and data visualization. There have been recent developments in networking
graphics applications, via fiberoptic cables, to the university's new
Northeast Parallel Architectures Center, located on campus. Interfacing the
IRIS with NPAC's Connection Machine will shortly open up new possibilities
for further (and deeper) explorations with fractals in the complex plane.
1. Oppenheimer, Peter from a presentation at Siggraph '87 and a paper in the
course notes: The Modeling of Natural Phenomena, page 55
2. Papathomas & Julesz "Animation with fractals from variations on the
Mandelbrot set" The Visual Computer 1987, pages 23-25 (the deformations of the
set described in this article were done by varying the exponent in the function.)