© Joel Hagen
Monsters lurked in the domain of classical geometry at the dawn of the
20th century. Mathematicians expressed intellectual horror as they created
shapes of finite area bounded by infinite curves without tangents. The
orderly world views of Euclid and Newton were being challenged in the sciences
as the esthetics of structure and harmony were being challenged in the
arts. The nature of disorder was embraced and explored, yielding some of
the most powerful movements in the evolution of the arts and sciences.
Fractal mathematics is an offspring of that exploration and has attracted
the attention of artists and scientists alike.
This column will suggest some lines of exploration
you might try with any fractal program. I used Fractal Pro, a solid full-featured
assembly language program for Amiga with extensive AREXX support, ideal
for production work and animation.
One of the key concepts of fractal mathematics is that dimensions are not
identified by the simple integers of Euclid. For example, a line scribbled
densely on a sheet of paper is not one dimensional as classical geometry
would define it. That line tends to fill much of a two dimensional plane.
In his pioneering work on fractal mathematics, Benoit Mandelbrot suggests
that it is more accurate and useful to define the dimension of such a line
as a fraction greater than one and less than two (hence "fractal"
from the same root as "fraction"). This is far more than a philosophical
observation since fractal mathematics provides a means to precisely define
the dimension of that line as, for example, 1.1815. It thus becomes possible
to meaningfully measure and compare the relative irregularity of structures.
This capability is being exploited by researchers in a wide range of disciplines
from economics to geography to psychology.
The methods of fractal mathematics allow us to generate forms as well as
to analyze them. It is this aspect of the discipline that is most familiar
to computer artists. The computer can create images by assigning values
to points within regions of sets of numbers. The nature of these sets of
numbers is such that one can look deeper and deeper into any region and
see new structure revealed. At an infinity of scales new forms exist yet
share similarities. In fractal software, this exploration is typically
done by outlining a region in a generated image then zooming in or magnifying
that region to reveal new detail. This sort of exploration can be done
out of sheer curiosity, but it can also be done with specific artistic
goals in mind.
In the illustration, image #1 was generated using Fractal Pro's Mandelbrot
SineM button. Zooming in on the area outlined in white produced the basis
for image #2. I was looking for parallel lines and shapes with an organic
feel, so my choices were not accidental. It only took one level of magnification
to find a suitable region. Working at Fractal Pro's 24 bit level, green
shading was applied before saving.
The palette of that image was further changed with ADPro's balancing controls
and histogram equalization as it moved through a series of experiments.
By this time, I was thinking of it as a potential texture for mapping to
a 3D object. With that in mind, a ten pixel motion blur was applied in
ImageFX along the direction of the banding to enhance its soft linearity.
The result is what you see in image #2.
That 24 bit IFF image was loaded into Caligari's Texture maker in the Attributes
area of the BRender interface where it was converted into a Caligari format
texture. It was placed on a sphere using the Default mapping button and
Phong shading with shininess raised to let two local lights create soft
surface reflections. Careful rotation of the point of view concealed the
seam where texture edges met. The result is image #4. Similar fractal images
could have good potential as stone surfaces or even as banded gas giant
planets depending on what surface attributes you choose to apply in a 3D
program.
My original reason for looking for parallel structures in a fractal
image had nothing to do with 3D textures, but it is easy to become sidetracked.
I had set out hoping to discover a source of unique terrain for landscape
generation. As beautiful and convincing as are the random landscapes produced
by Vista Pro and Scenery Animator, they lack diversity. They are typically
an undulation of peaks and depressions that could be roughly characterized
as conical. Visually, they are good approximations of much of the mountainous
terrain of the Rockies and the Sierras, however I have always been intrigued
by the look of the Appalachians of the Eastern United States. The geologic
processes that folded these parallel wrinkles on the passive eastern continental
margin are different from the processes that raise the Sierras from the
colliding American and Pacific crustal plates. If you have a chance, look
at U.S. Geological Survey map I-2206 which is one of the largest shaded
relief maps of the United States ever produced. It employs computer generated
shading of Digital Elevation Model (DEM) data gathered from standard contour
maps. The detail is incredible, and the variety of terrain is tantalizing.
To acquire this map, contact your closest USGS office, it costs about five
dollars.
Fractal Pro has options which can automatically save any fractal image
as a DEM file as well as an IFF image. These DEMs can be loaded directly
into Vista Pro or Scenery Animator and generated as landscapes. The DEM
file for image #2 (before all my processing) produced the landscape in
image #3 when rendered in Vista Pro from the vantage shown by the arrow.
Given the self-similar nature of fractal images cascading recursively into
infinity, I have probably found a source for all the Appalachian ridges
I will ever want to render. Now if I could just find the right area of
that Julia set to uncover some lunar sinuous rilles.
If you haven't explored fractals, you are missing something fascinating
that goes far beyond colorful pictures. You may be missing an opportunity
to shift your perception of nature and challenge your concept of dimension
and reality. Fractal mathematics is not an ultimate truth about the structure
of nature. It is a tool that has given rise to tools in a quest for truth.
Fractals, chaos, and other mathematical monstrosities all provide science
some basic mechanisms to accept a subjective universe and examine it meaningfully
with subjective measurement. This might be looked upon as part of the "Easting"
of the West. That piece of paper with the scribbled line can be viewed
from such a distance that it has zero dimension. Viewed closer, it might
appear a uniform two dimensional grey plane. Closer still the apparently
one dimensional line is visible and invites measurement, but at what scale?
The line is really particles of graphite. Measure around and over each
particle? Each molecule? What is the reality of that line? What is its
real length and what dimension does it occupy?
These are not idle questions, not to the western astrophysicist nor the
eastern monk nor the African artist... their minds are focused on the same
universe.