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Julia set

Julia sets, described by Gaston Julia, are fractal shapes defined on the complex number plane. Given two complex numbers, c and z, we define the following recursion:

zn+1 = zn2 + c

For a given value of c, the Julia set consists of all values of z for which z, when iterated, does not "blow up" or tends to infinity. Julia sets are closely related to the Mandelbrot set which is the set of all values of c for which z=0+0i does not tend to infinity through application of the recursion.

The Mandelbrot set is, in a way, an index of all Julia sets, For any point on the complex plane (which represents a value of c) a corresponding Julia set can be drawn. We can imagine a movie of a point moving about the complex plane with its corresponding Julia set. When the point lies in the Mandelbrot set the Julia set is "all in one piece" or topologically unified. As the point crosses the boundary of the Mandelbrot set, the Julia set shatters into a Cantor dust of unconnected points.

If c is on the boundary of the Mandelbrot set, and is not a waist, the Julia set of c looks like the Mandelbrot set in sufficiently small neighborhoods of c. For instance:

  • At c=1/4, the cusp at the set's mouth, the Julia set outline is a closed curve with cusps all around.
  • At c=i, the shorter, front toe of the forefoot, the Julia set looks like a branched lightning bolt.
  • At c=-2, the tip of the long spiky tail, the Julia set is a straight line segment.
  • -3/4, 1/4+i/2, and e2πi/5/2-e4πi/5/4 (-0.482-0.532i) are waists, The Julia set at c=-3/4 actually does look like the Mandelbrot set there, but the other two do not.



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Julia set

... a value of c) a corresponding Julia set can be drawn. We can imagine a movie of a point moving about the complex plane with its corresponding Julia set. When the point ...