Scott Burns creates digital imagery based on mathematics. He writes short
computer programs that generate this imagery, and includes the source code that
generated the image on the back of each piece. He refers to this work as "open
source artwork." The organic shapes and patterns displayed in this artwork exist
naturally in the mathematics, but take on visual form only with the aid of
computer graphics.1
To understand the process, think of the image as comprising a large number of
tiny pixels. The goal is to color each pixel according to the outcome of a
single, simple iterative process. The x-y coordinates of each pixel provide the
starting point for the iterative process. This starting value (expressed as a
coordinate pair, or as a single complex number) is entered into a mathematical
expression, which generates a new value. The value obtained from the expression
is entered back into the same expression, producing yet another value. This
process repeats until the value satisfies some mathematical condition defined by
the artist (called an orbit trap), at which point the iteration terminates. The
pixel is then colored according to a color lookup table, with the aid of a
transfer function that selects a color according to some characteristic of the
outcome of the iteration. At this point, one pixel has been colored. The same
expression is used for each and every pixel of the image, differing only in the
starting value used to initiate the process. This is then repeated sixty million
times, once for each of the remaining pixels, to produce the completed image.
There is no manipulation of the image once it has been created by the
computer program--no hand coloring or drawing. Control of the technique is
achieved through the artist's selection and manipulation of the mathematical
equation, the orbit traps, and the color mapping that relates terminating events
to pixel colors. The artist's manipulations of the code tend to be global in
nature, as compared to the local manipulations one achieves with a paintbrush.
Scott's main artistic skill is his ability to deliberately produce a desired
outcome through these code-based manipulations, in contrast to typical "fractal
artists," who tend to rely on random exploration to stumble upon something that
is eye catching.
The text of the computer program that implements the above process is
included on the back of each piece, or in some cases, is printed on the bottom
of the larger pieces. It is written in a generic computer language that is
straightforward to translate to any of the commonly used programming languages.
The final print is produced on a wide format printer (Epson 7600), using
archival quality inks and papers. The work is offered in signed and numbered
limited editions, using acid free materials for matting and framing.
Q: What is an algorithm?
A: According to Wikipedia: "an algorithm is a finite list
of well-defined instructions for accomplishing some task that, given an
initial state, will terminate in a defined end-state." All of this artwork is
created by algorithms that I design and implement in very short computer
programs, which are printed on the back of each piece.
Q: Do you
draw, paint, or color them by hand?
A: No, the
images that you see are exactly how they appear on the screen when I run the
program. If I wish to modify an image, I modify the code, not the image
directly.
Q: What
software program do you use?
A: The code
you see on the back of the artwork is the text of the entire computer program.
It generates a bitmapped image file, similar to the file that your digital
camera makes. There is no other computer program that I use. I use no
"rendering" software, nor do I "Photoshop" the image.
Q: What
programming language is that on the back?
A: It is a
combination of C, Basic, and Fortran. I took the parts of each language that I
thought were easiest to understand, and combined them into a custom language.
If you wish to program it yourself, you will have to translate the code into a
single language of your choice.
Q: Can I
really create these images from the code printed on the back?
A: Yes!
However, you will need to know how to program a computer. That is different
than knowing how to run programs on a computer.
Q: How are
they printed?
A: They are
printed on an Epson 7600 wide format printer, using archival papers and ink
sets. Independent tests indicate there should be no noticeable fading for at
least 70 years. They are matted with acid free materials.
Q: How did
you develop this technique?
A: It grew
out of my research at the University of Illinois in the mid '80s. I was
studying the behavior of iterative numerical methods in engineering design, in
support of a National Science Foundation grant, and started seeing these
interesting shapes and colors. I then decided to pursue it as an art form, and
have been exhibiting since 1990.
Q: Is this what you do for a living?
A: No, I teach engineering as my "day job." This is just
a hobby that has gotten way out of hand!
Q: How long
does it take to create one of these?
A: Back in
the '80s, it took a long time just to run the program. I had one that ran for
24 days! Today’s fast computers can do the computation in a couple of hours
and I do all my work on a laptop. Now, the most time consuming step is
designing and modifying the mathematical expressions to achieve a desired
graphical outcome. I have control of the main features of the image (examples
below). Nevertheless the finer details are often a big surprise.
Q: Are these fractals?
A: Yes, these are fractals of my own design. In contrast
to "standard" fractals, such as the Mandelbrot set or the Julia set, these
fractals are custom designed to express my artistic intent more fully.
Q: What is a fractal?
A: The simplest explanation I can offer is that a fractal
is a geometric form that exhibits special characteristics, including a self
similarity at all levels of magnification. Can you find an example of this
when comparing images 51 and 79?
Q: Do you have control of the shapes and patterns?
A: Yes, to a large degree. Now that I have been doing
this for 25 or so years, I can deliberately produce desired shapes and
patterns from my understanding of how recurrences and orbit traps will behave.
For example, design #80 was inspired by the interesting patterns I saw on this
palm tree while on vacation in Florida.
I designed equations to mimic these patterns.
Another example is design #84. I intentionally designed
mathematical expressions to produce the woven features of this image.
Q: How are the colors selected?
A: First, I define a color palette and implement it in
the code as a color lookup table. Then I instruct the computer how to select
colors from the palette when orbits are trapped while executing a recurrence
(see later for explanation of these terms). The hue, saturation, and value are
all mathematically determined by the algorithm I create.
Q: What is a recurrence?
A: This is short for "recurrence relation," which is a
mathematical expression that produces a new value of a sequence from previous
values. For example, the recurrence xn+1 = xn2
produces the sequence 2, 4, 16, 256, . . . when initiated with x = 2.
Q: What is an orbit trap?
A: An orbit trap is a mathematical condition that
determines when a recurrence should terminate. For example, using the
recurrence xn+1 = xn2, the orbit trap x>100
will terminate the recurrence on the third iteration when starting with x = 2.
Q: How does the orbit trap determine the color?
A: I define the orbit traps to capture different
behaviors of the recurrence. For example, xn+1 = xn2
has three distinct behaviors, according to whether the starting point is
greater than 1 (it blows up to infinity), equal to 0 or 1 (it stays the same),
or between 0 and 1 (it converges to zero). These behaviors can be captured by
different orbit traps. Different colors can then be assigned to each orbit
trap according to a wide variety of criteria, such as the iteration count at
the time of capture, or the location within the orbit trap where the iterate
lands.
Q: How do you get a 2-D image from a single variable?
A: The key is to work in the complex plane. The symbol z
that you see in the formulas represents a complex number—one which has a
"real" part and an "imaginary" part. Each pixel in the image has a unique x-y
coordinate pair, which can be used to represent a unique z value. The
recurrence is a function of a single complex variable, which generates a
sequence of complex numbers. The orbit trap is also expressed in terms of the
complex variable. Thus, the mathematical treatment applied to each pixel is
identical, except for the starting value, which is a the z value corresponding
to each pixel.
Q: You seem to use Newton's method a lot. Why?
A: Newton's method has long been known to exhibit chaotic
behavior when applied to certain equations. Although it converges quickly when
in the vicinity of a solution, it can jump violently in early iterations. This
was noted by mathematicians in the 1800s, but the aesthetic quality of this
behavior was not appreciated until computer graphics revealed it visually.
Q: How do you discern geeks from non-geeks?
A: The non-geeks are fascinated by the colored side of
the artwork. The geeks think the art is the computer code posted on the back
of the artwork! :)
