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.

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" 0.004 minimum distance between points "Bound" 6 radius 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)

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 :

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:

Wolfram Research (2014), JuliaSetPoints, Wolfram Language function, https://reference.wolfram.com/language/ref/JuliaSetPoints.html.

Text

Wolfram Research (2014), JuliaSetPoints, Wolfram Language function, https://reference.wolfram.com/language/ref/JuliaSetPoints.html.

CMS

Wolfram Language. 2014. "JuliaSetPoints." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/JuliaSetPoints.html.

APA

Wolfram Language. (2014). JuliaSetPoints. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/JuliaSetPoints.html

BibTeX

@misc{reference.wolfram_2022_juliasetpoints, author="Wolfram Research", title="{JuliaSetPoints}", year="2014", howpublished="\url{https://reference.wolfram.com/language/ref/JuliaSetPoints.html}", note=[Accessed: 05-June-2023 ]}

BibLaTeX

@online{reference.wolfram_2022_juliasetpoints, organization={Wolfram Research}, title={JuliaSetPoints}, year={2014}, url={https://reference.wolfram.com/language/ref/JuliaSetPoints.html}, note=[Accessed: 05-June-2023 ]}