Maxius' Miracle

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Consider a complex iteration function, f(z)=z^2+c, where z and c are complex number, respectively.

Julia Set: a plot in the complex plane of initial variables

If we fix c as a constant number, the complex ineration function f(z) will convergence or divergence according to different initial variables z_0. We can visualization the divergence "speed" in a complex plane of initial variables, because z_0 is a complex number. For differnt c, we will get differnet Julia sets. Different divergence speeds are labeled with different colors.

Mandelbrot Set: a plot in the complex plane of parameters

If we fix initial value z_0 as a constant number, the complex ineration function f(z) will convergence or divergence according to diffenent paramter c. We can visualization the divergence "speed" in a complex plane of parameter, because c is a complex number. If z_0 is set as zero, we will get the Mandelbrot set. Different divergence speeds are labeled with different colors. An important therom for Mandelbrot set is that the complex iteration function will convergence if and only if every absolute value of iterated value is less than 2.

Two interesting introductions to Julia set and Mandelbrot set can be found at:

Link_1, Link_2.

More comprehensive reviews can be found at Wikipedia or WolframMathWorld

Julia Set Wiki Wolfram

Mandelbrot Set Wiki Wolfram

The following images are Julia sets with different c.

c = 0.45, -0.1428
  

c = 0.285, 0.01
  

c = 0.285, 0
  

c = -0.8, 0.156
  

c = -0.835, -0.2321
  

c = -0.70176, -0.3842
  

This image is a Mandelbrot set.