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Wolfram Language & System Documentation Center
JuliaSetPlot
  • See Also
    • MandelbrotSetPlot
    • JuliaSetIterationCount
    • ListPlot
    • ArrayPlot
    • DensityPlot
    • ContourPlot
  • Related Guides
    • Iterated Maps & Fractals
    • Complex Visualization
    • See Also
      • MandelbrotSetPlot
      • JuliaSetIterationCount
      • ListPlot
      • ArrayPlot
      • DensityPlot
      • ContourPlot
    • Related Guides
      • Iterated Maps & Fractals
      • Complex Visualization

JuliaSetPlot[f,z]

plots the Julia set of the rational function f of the variable z.

JuliaSetPlot[c]

plots the Julia set of the function .

Details and Options
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Options  
AspectRatio  
Axes  
AxesLabel  
Show More Show More
AxesOrigin  
AxesStyle  
Background  
ColorFunction  
Frame  
FrameLabel  
FrameStyle  
FrameTicks  
FrameTicksStyle  
ImageResolution  
ImageSize  
PerformanceGoal  
PlotLegends  
PlotRange  
PlotStyle  
PlotTheme  
Ticks  
TicksStyle  
Possible Issues  
Neat Examples  
See Also
Related Guides
History
Cite this Page
BUILT-IN SYMBOL
  • See Also
    • MandelbrotSetPlot
    • JuliaSetIterationCount
    • ListPlot
    • ArrayPlot
    • DensityPlot
    • ContourPlot
  • Related Guides
    • Iterated Maps & Fractals
    • Complex Visualization
    • See Also
      • MandelbrotSetPlot
      • JuliaSetIterationCount
      • ListPlot
      • ArrayPlot
      • DensityPlot
      • ContourPlot
    • Related Guides
      • Iterated Maps & Fractals
      • Complex Visualization

JuliaSetPlot

JuliaSetPlot[f,z]

plots the Julia set of the rational function f of the variable z.

JuliaSetPlot[c]

plots 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.
  • JuliaSetPlot has the same options as Graphics, with the following additions and changes: [List of all options]
  • AspectRatio Automaticratio of height to width of the plot
    Axes Falsewhether to include axes
    ColorFunction Automatichow to color the exterior
    ColorFunctionScaling Truewhether to scale arguments to ColorFunction
    Frame Truewhether to include a frame
    ImageResolution Automaticresolution of points in the Julia set
    MaxIterationsAutomatichow many iterations to allow for each point
    MethodAutomaticthe method to generate the image
    PerformanceGoal $PerformanceGoalaspects of performance to try to optimize
    PlotLegends Nonelegends for the number of interations
    PlotRange Automaticrange of values to include
    PlotRangeClippingTruewhether to clip at the plot range
    PlotStyle Automaticgraphics directives to specify the style for each point
    PlotTheme $PlotThemetheme to use for styles and appearances
  • The possible methods are "InverseIteration", "EscapeTime", and "OrbitDetection".
  • With Method->"InverseIteration", JuliaSetPlot produces a ListPlot showing individual points of the Julia set.
  • With Method->"EscapeTime" or with Method->"OrbitDetection", JuliaSetPlot produces a Graphics object containing a Raster primitive.
  • With ColorFunction->f, where f is a function, the argument of f is a real number in proportional to the number of iterates, and f must return color directives, such as RGBColor and Hue, or named colors, such as Red and Blue.
  • ColorFunction->"name" is equivalent to ColorFunction->(If[#1,Black,ColorData["name"][#]]&).
  • The list of possible color function names is given by ColorData["Gradients"].
  • With ColorFunctionNone, the default color used by PlotStyle is given by ColorData["DefaultPlotColors",1].
  • List of all options

    • AlignmentPointCenterthe default point in the graphic to align with
      AspectRatioAutomaticratio of height to width of the plot
      AxesFalsewhether to include axes
      AxesLabelNoneaxes labels
      AxesOriginAutomaticwhere axes should cross
      AxesStyle{}style specifications for the axes
      BackgroundNonebackground color for the plot
      BaselinePositionAutomatichow to align with a surrounding text baseline
      BaseStyle{}base style specifications for the graphic
      ColorFunctionAutomatichow to color the exterior
      ColorFunctionScalingTruewhether to scale arguments to ColorFunction
      ContentSelectableAutomaticwhether to allow contents to be selected
      CoordinatesToolOptionsAutomaticdetailed behavior of the coordinates tool
      Epilog{}primitives rendered after the main plot
      FormatTypeTraditionalFormthe default format type for text
      FrameTruewhether to include a frame
      FrameLabelNoneframe labels
      FrameStyle{}style specifications for the frame
      FrameTicksAutomaticframe ticks
      FrameTicksStyle{}style specifications for frame ticks
      GridLinesNonegrid lines to draw
      GridLinesStyle{}style specifications for grid lines
      ImageMargins0.the margins to leave around the graphic
      ImagePaddingAllwhat extra padding to allow for labels etc.
      ImageResolutionAutomaticresolution of points in the Julia set
      ImageSizeAutomaticthe absolute size at which to render the graphic
      LabelStyle{}style specifications for labels
      MaxIterationsAutomatichow many iterations to allow for each point
      MethodAutomaticthe method to generate the image
      PerformanceGoal$PerformanceGoalaspects of performance to try to optimize
      PlotLabelNonean overall label for the plot
      PlotLegendsNonelegends for the number of interations
      PlotRangeAutomaticrange of values to include
      PlotRangeClippingTruewhether to clip at the plot range
      PlotRangePaddingAutomatichow much to pad the range of values
      PlotRegionAutomaticthe final display region to be filled
      PlotStyleAutomaticgraphics directives to specify the style for each point
      PlotTheme$PlotThemetheme to use for styles and appearances
      PreserveImageOptionsAutomaticwhether to preserve image options when displaying new versions of the same graphic
      Prolog{}primitives rendered before the main plot
      RotateLabelTruewhether to rotate y labels on the frame
      TicksAutomaticaxes ticks
      TicksStyle{}style specifications for axes ticks

Examples

open all close all

Basic Examples  (4)

Generate the filled Julia set of :

Generate the Julia set:

Show the filled Julia set with the points in the Julia set:

Show a legend of the number of iterations:

Scope  (7)

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

Generate the Julia set of a polynomial:

Generate the filled Julia set of a rational function:

Change the color function:

Show the points in the Julia set:

Show just the points:

Use a theme with simple ticks in a high-contrast color scheme:

Options  (71)

AspectRatio  (4)

By default, the ratio of the height to width for the plot is determined automatically:

Use a numerical value to specify the height-to-width ratio:

Make the height the same as the width with AspectRatio1:

AspectRatioFull adjusts the height and width to tightly fit inside other constructs:

Axes  (3)

By default, JuliaSetPlot uses a frame instead of axes:

Use axes instead of a frame:

Turn on each axis individually:

AxesLabel  (3)

No axes labels are drawn by default:

Place a label on the axis:

Specify axes labels:

AxesOrigin  (2)

The positions of the axes are determined automatically:

Specify an explicit origin for the axes:

AxesStyle  (4)

Change the style for the axes:

Specify the style of each axis:

Use different styles for the ticks and the axes:

Use different styles for the labels and the axes:

Background  (1)

Use a black background as contrast with the points in the Julia set:

ColorFunction  (3)

Use a built-in color function to color by a scaled number of iterations:

Zoom in to a region to see more details:

Use a named color gradient:

Write a custom color function:

Frame  (4)

JuliaSetPlot uses a frame by default:

Use FrameFalse to turn off the frame:

Draw a frame on the left and right edges:

Draw a frame on the left and bottom edges:

FrameLabel  (4)

Place a label along the bottom frame of a plot:

Frame labels are placed on the bottom and left frame edges by default:

Place labels on each of the edges in the frame:

Use a customized style for both labels and frame tick labels:

FrameStyle  (2)

Specify the style of the frame:

Specify the style for each frame edge:

FrameTicks  (9)

Frame ticks are placed automatically by default:

Use a frame with no ticks:

Use frame ticks on the bottom edge:

By default, the top and right edges have tick marks but no tick labels:

Use All to include tick labels on all edges:

Place tick marks at specific positions:

Draw frame tick marks at the specified positions with specific labels:

Specify the lengths for tick marks as a fraction of the graphics size:

Use different sizes in the positive and negative directions for each tick mark:

Specify a style for each frame tick:

Construct a function that places frame ticks at the midpoint and extremes of the frame edge:

FrameTicksStyle  (3)

By default, the frame ticks and frame tick labels use the same styles as the frame:

Specify an overall style for the ticks, including the labels:

Use different style for the different frame edges:

ImageResolution  (1)

Increase ImageResolution for finer plots:

ImageSize  (7)

Use named sizes such as Tiny, Small, Medium and Large:

Specify the width of the plot:

Specify the height of the plot:

Allow the width and height to be up to a certain size:

Specify the width and height for a graphic, padding with space if necessary:

Setting AspectRatioFull will fill the available space:

Use maximum sizes for the width and height:

Use ImageSizeFull to fill the available space in an object:

Specify the image size as a fraction of the available space:

PerformanceGoal  (2)

Generate a higher-quality plot:

Emphasize performance, possibly at the cost of quality:

PlotLegends  (1)

Legends show the number of iterations:

PlotRange  (2)

The plot range is chosen automatically, based on the Julia set:

Zoom in to a region to see more details:

PlotStyle  (2)

By default, the points are not shown unless ColorFunction->None:

Specify the plot style:

PlotTheme  (1)

Use a theme with increased contrast and axes:

Ticks  (9)

Ticks are placed automatically in each plot:

Use TicksNone to draw axes without any tick marks:

Use ticks on the axis, but not the axis:

Place tick marks at specific positions:

Draw tick marks at the specified positions with the specified labels:

Use specific ticks on one axis and automatic ticks on the other:

Specify the lengths for ticks as a fraction of the graphics size:

Use different sizes in the positive and negative directions for each tick:

Specify a style for each tick:

Construct a function that places ticks at the midpoint and extremes of the axis:

TicksStyle  (4)

By default, the ticks and tick labels use the same style as the axis:

Specify an overall tick style, including the tick labels:

Specify the tick style for each of the axes:

Use a different style for the tick labels and tick marks:

Possible Issues  (4)

JuliaSetPlot only works with rational functions:

When using orbit detection, sometimes no attractive orbits are detected:

If the value of the "Bound" method 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:

Neat Examples  (1)

Colorize the components of a Julia set:

See Also

MandelbrotSetPlot  JuliaSetIterationCount  ListPlot  ArrayPlot  DensityPlot  ContourPlot

Related Guides

    ▪
  • Iterated Maps & Fractals
  • ▪
  • Complex Visualization

History

Introduced in 2014 (10.0)

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

Text

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

CMS

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

APA

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

BibTeX

@misc{reference.wolfram_2025_juliasetplot, author="Wolfram Research", title="{JuliaSetPlot}", year="2014", howpublished="\url{https://reference.wolfram.com/language/ref/JuliaSetPlot.html}", note=[Accessed: 01-May-2026]}

BibLaTeX

@online{reference.wolfram_2025_juliasetplot, organization={Wolfram Research}, title={JuliaSetPlot}, year={2014}, url={https://reference.wolfram.com/language/ref/JuliaSetPlot.html}, note=[Accessed: 01-May-2026]}

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