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Magnified Views of the Mandelbrot Set

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center
magnification
1
2
4
8
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32
The Mandelbrot set is defined as the collection of points
c
in the complex plane for which the orbit of the iterated relation
z
n+1
=
2
z
n
+c
, known as the Julia set for
c
, remains in a finite, simply connected region as
n∞
. 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
|
z
n
|≥2
for any value of
n
, 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
|z|≥2
. 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
5
2
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.
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