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Mandelbrot Set: Complex Plane
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The whole Mandelbrot set lies within a circle of radius 2.5 centered at the origin of the complex plane. Although finite in area, the Set has a boundary that is infinitely long and has a Hausdorff dimension of 2. The overall appearance of the Mandelbrot set is that of a series of disks. These disks have irregular borders and decrease in size heading out along the negative real axis; ... the ratio of the diameter of one disk to the next approaches a constant. More complex shapes branch out from the disks. One region of the Mandelbrot set containing spiral shapes is known as Seahorse Valley because it resembles a seahorse's tail.
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The Mandelbrot set (M) has been called the most complex object in mathematics, and continues to be the subject of active research. One open question is, what is the area of M? It is well known that the set is bounded by a circle of radius 2, centered at the origin of the complex plane. Thus, the area is certainly less than 4p, or approximately 12.6. Indeed, the area is much less than that. The left-most extent of the set ends with the spike at x = -2, and the right side extends out to approximately x = 0.47.
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Douady and Hubbard made a major contribution to the understanding of the Mandelbrot set and its role in the description of dynamics of iterative processes. As the Fundamental Theorem of Algebra clearly indicates, the complex plane rather than the real line is the proper place for the study of polynomials. And, in hindsight, the study of even simple polynomials of degree 2 in the complex plain has thrown a new light on the behavior of more general iterative processes.
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