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12.7: Special Topic- Exploring the Mandelbrot Set

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    The Mandelbrot set is one of the most fascinating fractals. It is named after its discover, IBM mathematician Benoit Mandelbrot. The Mandelbrot set itself is the black, heart-shaped image shown in Figure [fig-mandel]. What makes the Mandelbrot set so interesting is that with the help of a Java GUI you can explore the set as if you were taking a trip through outer space. The most interesting regions to explore are those just along the boundary of the set. For example, notice that the boundary contains numerous circular shapes, each of which is itself studded with circular shapes. This is an example of the scaled self-similarity that we found to be so prevalent in recursive structures. By continually expanding the regions around the boundary, you’ll find an infinite recursion of fascinating images and shapes. In some regions of the set you’ll even find miniature replications of the set itself.

    The Mandelbrot set is generated by an iterated function system. The mathematics underlying this fascinating object is quite accessible, and there are a number of online tutorials that explain how the set is generated and how the pictures are produced. Many of the Mandelbrot and fractal Web sites contain excellent Java applets that let you explore the Mandelbrot set as well as related sets. An excellent place to start your exploration would be David Joyce’s award-winning Web site,

    http://aleph0.clarku.edu/~djoyce/julia/

    which contains references to a number of other good sites. For a tutorial on how the various Mandelbrot set-generating Java programs work, see

    http://storm.shodor.org/mteach/

    This page titled 12.7: Special Topic- Exploring the Mandelbrot Set is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Ralph Morelli & Ralph Wade via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.