 
Excerpt from my Final Document, "Using
Emergent Design to Implement ComputerEnabled Learning Environments in K12 Public Schools"
© Copyright 2002 by Carol
CaldwellEdmonds, all
rights reserved
The science of
complexity involves investigations of how complex phenomena arise from simple components
and
simple
interactions (Resnick and Wilensky 1996). These
investigations began when computers were first used in
scientific
research to model weather systems. Graphing
certain mathematical problems yielded unexpected
results.
Although the details of these numerous investigations are beyond the scope of this paper,
they are well
described in
James Gleick’s book Chaos (1987). Gleick likens the effect computers have had
on recent
scientists’
work to the effect microscopes had on previous generations of scientists’ work. Until the computer
was invented,
there was no way to perform millions of calculations in seconds. When computers were used to
calculate and
graph the results of millions of repetitions of certain math problems, a whole new science
began
emerging. New things could be seen.
Figure 1. a, b, c Images of a Mandelbrot Set 



Figure 1.a
These images are examples of graphing
the results of a math problem. Before computers with graphics capabilities, these images
were unknown. In 1979, Benoit Mandelbrot used
huge IBM mainframe computers to generate the first crude black and white images of this
set, now named for him (Gleick 1987, 223). The program I used to create these images is
available on the Internet for $20.00. It is “careware,” software created by a
group of programmers to raise money for helping the education of autistic children. Phillip Crews, the creator of the program, said
the children helped by the foundation use the program and create images to color. (Email correspondence) 
Figure 1. a, b, c Images of a Mandelbrot Set 
Figure 1.b 


Zooming in on the area I outlined in red
(Figure 1.a) above creates this second image. Programs
that create images like this are surprisingly brief because they use recursion. Each point on this image was the result of
a mathematical calculation from university level mathematics. Yet, the whole image pops on the screen in a
fraction of a second. Since the “zooming
in” theoretically never ends, the Mandelbrot set provides an image of an infinite
process. 
Figure 1. a, b, c Images of a Mandelbrot Set 
Figure 1.c

This image is the
result of zooming in on the area outlined by the red square in the previous image above
(Figure 1.b) seven times. Mandelbrot investigated the set after
years of thinking about the irregularities of markets, coastlines, and other behavior that
the standard, neat textbook math problems dismissed as exceptions or anomalies. These images became the symbol of the science
of Chaos; the fringes of the set are areas where the results of orderly calculations
become chaotic. The white area is the set;
the colors are made by points outside of the set, colorcoded by how many times the
recursive program executes calculations for the point it plots up to a preset limit
(usually related to how many decimal places a computer can handle) 
The images of the Mandelbrot Set
literally illustrate how computers were used to reveal patterns and the drift away from
neat patterns. In the images, zooming in and
recreating a new image (i.e., starting a new set of recursive instructions on a smaller
scale, as in the treedrawing program example) reveal patterns in the “edges” of
the set that resemble the original large set. The
patterns appear again as one continues “zooming in.” The same (white) shape is distorted but present in
the fringes of the last image. Mandelbrot
coined the term “fractals” in the 1970s to describe selfsimilar patterns that
emerge from certain recursive calculations (Gleick 1987, 98), but the Mandelbrot Set
revealed slight distortions of the original pattern in each recursivelyplotted image. Mandelbrot investigated patterns in statistical
studies applied to diverse areas such as stock market prices, the probabilities of words
occurring in English, and the behavior of fluids subjected to turbulence. As the use of computers increased in scientific
studies of complexity, researchers from different fields including physics, math, biology,
and medicine discovered more applications for the math and programs that had revealed the
behavior of points that make up the Mandelbrot Set (Gleick 1987).
Research into complex systems touches on
some of the deepest issues in science and philosophy—order vs. chaos, randomness vs.
determinacy, analysis vs. synthesis. In the
minds of many, the study of complexity is not just a new science, but a new way of
thinking about all science, a fundamental shift from the paradigms that have dominated
scientific thinking for the past 300 years (Resnick and Wilensky 1996).
These
comments describe the truly revolutionary impact of computers. 
. 
