JuliaSetPoints

JuliaSetPoints[f,z]

returns a list of coordinates approximating the real and imaginary parts of the complex numbers in the Julia set of the rational function f of the variable z.

JuliaSetPoints[c]

returns a list of coordinates of points approximating the Julia set of the function .

Details and Options

  • The Julia set of a function f is the closure of the set of all repelling fixed points of f.
  • JuliaSetPoints uses the same "InverseIteration" algorithm as JuliaSetPlot.
  • JuliaSetPoints has the options:
  • "ClosenessTolerance"0.004minimum distance between points
    "Bound"6radius around the origin in which to search
  • For polynomial functions, "Bound" is automatically determined to ensure the entire Julia set is captured.

Examples

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Basic Examples  (2)

Find points of the Julia set of :

Find points of the Julia set of :

Scope  (2)

JuliaSetPoints[c] generates the Julia set of a function of the form :

JuliaSetPoints[f,z] generates the Julia set of polynomials or more general rational functions:

Options  (2)

"ClosenessTolerance"  (1)

Increase "ClosenessTolerance" to make a quick, low-resolution picture of a Julia set:

Decrease "ClosenessTolerance" to make a high-resolution picture of a small part of a Julia set:

"Bound"  (1)

For some rational functions, increasing "Bound" can find more points:

Properties & Relations  (2)

JuliaSetPlot[c] generates essentially a ListPlot of the result of JuliaSetPoints[c]:

JuliaSetPoints[c] is the same as JuliaSetPoints[z^2+c,z]:

Possible Issues  (1)

If the value of the "Bound" option is too low for a rational function, no points may be returned:

Some very large Julia sets can take a long time to compute with this method:

Interactive Examples  (1)

Explore the Julia sets for which the parameter c is on the unit circle:

Neat Examples  (3)

Stack successively finer approximations to a Julia set:

Add a dimension by varying a parameter:

Visualize the Julia sets given by points on part of the unit circle:

Introduced in 2014
 (10.0)