The Mandelbrot Set
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Excerpt from my Final Document, "Using Emergent Design to Implement Computer-Enabled Learning Environments in K-12 Public Schools" Copyright 2002 by Carol Caldwell-Edmonds, 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, color-coded 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 tree-drawing 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 self-similar patterns that emerge from certain recursive calculations (Gleick 1987, 98), but the Mandelbrot Set revealed slight distortions of the original pattern in each recursively-plotted 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. 

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