Julia set

Julia sets, described by Gaston Julia, are fractal shapes defined on the complex number plane. Given an iterated map of the complex plane to itself, or a collection of such maps, the Julia set for this system can be defined (informally) as the set of points for which nearby points do not exhibit similar behaviour under repeated iterations of the map(s).

Relation to Mandelbrot set

Julia sets are closely related to the Mandelbrot set which is the set of all values of c for which zn = zn−12 + c does not tend to infinity through application of the recursion with z0 = 0. Like the Mandelbrot set, the Julia set is often plotted with different colors signifying the number of iterations carried out before the modulus of z becomes larger than 2.

Related Topics:
Mandelbrot set - Plot

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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 connected. Otherwise, the Julia set is a Cantor dust of unconnected points.

Related Topics:
Connected - Cantor dust

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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:

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• 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 e^2πi/5/2 − e^4πi/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.