Magnified Views of the Mandelbrot Set

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The Mandelbrot set is defined as the collection of points in the complex plane for which the orbit of the iterated relation , known as the Julia set for , remains in a finite, simply connected region as . The Mandelbrot set, which has a fractal boundary, is shown in black on the graphic. Two dominant features are a cardioid and a circular disc tangent to one another. If for any value of , then the point cannot belong to the Mandelbrot set. Such points fall in a colored region, with different colors determined by the number of steps it takes for an orbit to reach a value . Magnification of various regions of the complex plane reveals an incredible variety of fractal structures. The classic computer-generated images were published by Peitgen and Richter (The Beauty of Fractals: Heidelberg, Springer-Verlag, 1986). This Demonstration cannot reproduce their computational power, but a general idea of the richness of the Mandelbrot set can be obtained with magnifications up to with 300×300-pixel resolution. The 2D slider centers the image on a desired region for magnification. The magnification should be increased stepwise for optimal control.

Contributed by: S. M. Blinder (March 2011)
Open content licensed under CC BY-NC-SA



Snapshot 1: the region between the two main features is known as "seahorse valley"

Snapshot 2: each circular disk grows smaller circular disks; "baby Mandelbrot sets" also appear

Snapshot 3: one of many remarkably complex fractal patterns under higher magnification

A selection of Peitgen and Richter's images is shown in a Wikipedia article.

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