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Unformatted text preview: Chapter 10 Mandelbrot Set Fractals, topology, complex arithmetic and fascinating computer graphics. Benoit Mandelbrot is a Polish/French/American mathematician who has spent most of his career at the IBM Watson Research Center in Yorktown Heights, N.Y. He coined the term fractal and published a very influential book, The Fractal Ge ometry of Nature , in 1982. An image of the now famous Mandelbrot set appeared on the cover of Scientific American in 1985. This was about the time that computer graphical displays were first becoming widely available. Since then, the Mandelbrot set has stimulated deep research topics in mathematics and has also been the basis for an uncountable number of graphics projects, hardware demos, and Web pages. To get in the mood for the Mandelbrot set, consider the region in the complex plane consisting of the values z for which the trajectories defined by z k +1 = z 2 k , k = 0 , 1 ,... remain bounded at k → ∞ . It is easy to see that this set is simply the unit disc,  z  < = 1, shown in figure 10.1. If  z  < = 1, the repeated squares remain bounded. If  z  > 1, the repeated squares are unbounded. The boundary of the unit disc is the unit circle,  z  = 1. There is nothing very difficult or exciting here. The definition is the Mandelbrot set is only slightly more complicated. It involves repeatedly adding in the initial point. The Mandelbrot set is the region in the complex plane consisting of the values z for which the trajectories defined by z k +1 = z 2 k + z , k = 0 , 1 ,... remain bounded at k → ∞ . That’s it. That’s the entire definition. It’s amazing that such a simple definition can produce such fascinating complexity. Copyright c 2009 Cleve Moler Matlab R is a registered trademark of The MathWorks, Inc. TM August 8, 2009 1 2 Chapter 10. Mandelbrot Set1.510.5 0.5 1 1.51.510.5 0.5 1 1.5 Figure 10.1. The unit disc is shown in red. The boundary is simply the unit circle. There is no intricate fringe.21.510.5 0.5 11.510.5 0.5 1 1.5 Figure 10.2. The Mandelbrot set is shown in red. The fringe just outside the set, shown in black, is a region of rich structure. 3 Figure 10.3. Two trajectories. z0 = .25.54i generates a cycle of length four, while nearby z0 = .22.54i generates an unbounded trajectory. Figure 10.2 shows the overall geometry of the Mandelbrot set. However, this view does not have the resolution to show the richly detailed structure of the fringe just outside the boundary of the set. In fact, the set has tiny filaments reaching into the fringe region, even though the fringe appears to be solid black in the figure. It has recently been proved that the Mandelbrot set is mathematically connected, but the connected region is sometimes so thin that we cannot resolve it on a graphics screen or even compute it in a reasonable amount of time....
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 Spring '11
 Adams
 Topology, Complex number, Fractal, Mandelbrot set, Benoît Mandelbrot

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