--- title: "JuliaSetPlot" language: "en" type: "Symbol" summary: "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 f(z) == z^2 + c." keywords: - julia set - mandelbrot - complex dynamics canonical_url: "https://reference.wolfram.com/language/ref/JuliaSetPlot.html" source: "Wolfram Language Documentation" related_guides: - title: "Iterated Maps & Fractals" link: "https://reference.wolfram.com/language/guide/IteratedMapsAndFractals.en.md" - title: "Complex Visualization" link: "https://reference.wolfram.com/language/guide/ComplexVisualization.en.md" related_functions: - title: "MandelbrotSetPlot" link: "https://reference.wolfram.com/language/ref/MandelbrotSetPlot.en.md" - title: "JuliaSetIterationCount" link: "https://reference.wolfram.com/language/ref/JuliaSetIterationCount.en.md" - title: "ListPlot" link: "https://reference.wolfram.com/language/ref/ListPlot.en.md" - title: "ArrayPlot" link: "https://reference.wolfram.com/language/ref/ArrayPlot.en.md" - title: "DensityPlot" link: "https://reference.wolfram.com/language/ref/DensityPlot.en.md" - title: "ContourPlot" link: "https://reference.wolfram.com/language/ref/ContourPlot.en.md" --- # 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 $f(z)=z^2+c$. ## 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: [] | | | | | --------------------- | ----------------- | ------------------------------------------------------- | | AspectRatio | Automatic | ratio of height to width of the plot | | Axes | False | whether to include axes | | ColorFunction | Automatic | how to color the exterior | | ColorFunctionScaling | True | whether to scale arguments to ColorFunction | | Frame | True | whether to include a frame | | ImageResolution | Automatic | resolution of points in the Julia set | | MaxIterations | Automatic | how many iterations to allow for each point | | Method | Automatic | the method to generate the image | | PerformanceGoal | \$PerformanceGoal | aspects of performance to try to optimize | | PlotLegends | None | legends for the number of interations | | PlotRange | Automatic | range of values to include | | PlotRangeClipping | True | whether to clip at the plot range | | PlotStyle | Automatic | graphics directives to specify the style for each point | | PlotTheme | \$PlotTheme | theme 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 $[0,1]$ 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 | | | | | ---------------------- | ----------------- | ---------------------------------------------------------------------------------- | | AlignmentPoint | Center | the default point in the graphic to align with | | AspectRatio | Automatic | ratio of height to width of the plot | | Axes | False | whether to include axes | | AxesLabel | None | axes labels | | AxesOrigin | Automatic | where axes should cross | | AxesStyle | {} | style specifications for the axes | | Background | None | background color for the plot | | BaselinePosition | Automatic | how to align with a surrounding text baseline | | BaseStyle | {} | base style specifications for the graphic | | ColorFunction | Automatic | how to color the exterior | | ColorFunctionScaling | True | whether to scale arguments to ColorFunction | | ContentSelectable | Automatic | whether to allow contents to be selected | | CoordinatesToolOptions | Automatic | detailed behavior of the coordinates tool | | Epilog | {} | primitives rendered after the main plot | | FormatType | TraditionalForm | the default format type for text | | Frame | True | whether to include a frame | | FrameLabel | None | frame labels | | FrameStyle | {} | style specifications for the frame | | FrameTicks | Automatic | frame ticks | | FrameTicksStyle | {} | style specifications for frame ticks | | GridLines | None | grid lines to draw | | GridLinesStyle | {} | style specifications for grid lines | | ImageMargins | 0. | the margins to leave around the graphic | | ImagePadding | All | what extra padding to allow for labels etc. | | ImageResolution | Automatic | resolution of points in the Julia set | | ImageSize | Automatic | the absolute size at which to render the graphic | | LabelStyle | {} | style specifications for labels | | MaxIterations | Automatic | how many iterations to allow for each point | | Method | Automatic | the method to generate the image | | PerformanceGoal | \$PerformanceGoal | aspects of performance to try to optimize | | PlotLabel | None | an overall label for the plot | | PlotLegends | None | legends for the number of interations | | PlotRange | Automatic | range of values to include | | PlotRangeClipping | True | whether to clip at the plot range | | PlotRangePadding | Automatic | how much to pad the range of values | | PlotRegion | Automatic | the final display region to be filled | | PlotStyle | Automatic | graphics directives to specify the style for each point | | PlotTheme | \$PlotTheme | theme to use for styles and appearances | | PreserveImageOptions | Automatic | whether to preserve image options when displaying new versions of the same graphic | | Prolog | {} | primitives rendered before the main plot | | RotateLabel | True | whether to rotate y labels on the frame | | Ticks | Automatic | axes ticks | | TicksStyle | {} | style specifications for axes ticks | --- ## Examples (87) ### Basic Examples (4) Generate the filled Julia set of $z^2-1$ : ```wl In[1]:= JuliaSetPlot[-1] Out[1]= [image] ``` --- Generate the Julia set: ```wl In[1]:= JuliaSetPlot[-1, ColorFunction -> None] Out[1]= [image] ``` --- Show the filled Julia set with the points in the Julia set: ```wl In[1]:= JuliaSetPlot[-1, PlotStyle -> White] Out[1]= [image] ``` --- Show a legend of the number of iterations: ```wl In[1]:= JuliaSetPlot[0.365 - 0.37I, PlotLegends -> Automatic] Out[1]= [image] ``` ### Scope (7) ``JuliaSetPlot[c]`` generates the Julia set of a function of the form $z^2+c$ : ```wl In[1]:= JuliaSetPlot[-0.77 + 0.22I] Out[1]= [image] In[2]:= JuliaSetPlot[z^2 - 0.77 + 0.22I, z] Out[2]= [image] ``` --- Generate the Julia set of a polynomial: ```wl In[1]:= JuliaSetPlot[z^2 - z - 0.125, z] Out[1]= [image] ``` --- Generate the filled Julia set of a rational function: ```wl In[1]:= JuliaSetPlot[z ^ 2 / (z ^ 2 - 1), z] Out[1]= [image] ``` --- Change the color function: ```wl In[1]:= JuliaSetPlot[z^2 - 0.77 + 0.22I, z, ColorFunction -> Hue] Out[1]= [image] ``` --- Show the points in the Julia set: ```wl In[1]:= JuliaSetPlot[z^2 - 0.77 + 0.22I, z, ColorFunction -> Hue, PlotStyle -> Opacity[0.2, Black]] Out[1]= [image] ``` --- Show just the points: ```wl In[1]:= JuliaSetPlot[z^2 - 0.77 + 0.22I, z, ColorFunction -> None, PlotStyle -> Opacity[0.2, Black]] Out[1]= [image] ``` --- Use a theme with simple ticks in a high-contrast color scheme: ```wl In[1]:= JuliaSetPlot[-0.77 + 0.22I, PlotTheme -> "Web"] Out[1]= [image] ``` ### Options (71) #### AspectRatio (4) By default, the ratio of the height to width for the plot is determined automatically: ```wl In[1]:= JuliaSetPlot[z^2 - z - 1, z] Out[1]= [image] ``` --- Use a numerical value to specify the height-to-width ratio: ```wl In[1]:= JuliaSetPlot[z^2 - z - 1, z, AspectRatio -> 1 / 2] Out[1]= [image] ``` --- Make the height the same as the width with ``AspectRatio -> 1`` : ```wl In[1]:= JuliaSetPlot[z^2 - z - 1, z, AspectRatio -> 1] Out[1]= [image] ``` --- ``AspectRatio -> Full`` adjusts the height and width to tightly fit inside other constructs: ```wl In[1]:= plot = JuliaSetPlot[z^2 - z - 1, z, AspectRatio -> Full]; {Framed[Pane[plot, {75, 100}]], Framed[Pane[plot, {100, 100}]], Framed[Pane[plot, {100, 75}]]} Out[1]= {[image], [image], [image]} ``` #### Axes (3) By default, ``JuliaSetPlot`` uses a frame instead of axes: ```wl In[1]:= JuliaSetPlot[-0.77 + 0.22I] Out[1]= [image] ``` --- Use axes instead of a frame: ```wl In[1]:= JuliaSetPlot[-0.77 + 0.22I, Frame -> False, Axes -> True] Out[1]= [image] ``` --- Turn on each axis individually: ```wl In[1]:= {JuliaSetPlot[-0.77 + 0.22I, Frame -> False, Axes -> {True, False}], JuliaSetPlot[-0.77 + 0.22I, Frame -> False, Axes -> {False, True}]} Out[1]= {[image], [image]} ``` #### AxesLabel (3) No axes labels are drawn by default: ```wl In[1]:= JuliaSetPlot[-0.77 + 0.22I, Frame -> False, Axes -> True] Out[1]= [image] ``` --- Place a label on the $y$ axis: ```wl In[1]:= JuliaSetPlot[-0.77 + 0.22I, Frame -> False, Axes -> True, AxesLabel -> y] Out[1]= [image] ``` --- Specify axes labels: ```wl In[1]:= JuliaSetPlot[-0.77 + 0.22I, Frame -> False, Axes -> True, AxesLabel -> {x, y}] Out[1]= [image] ``` #### AxesOrigin (2) The positions of the axes are determined automatically: ```wl In[1]:= JuliaSetPlot[-0.77 + 0.22I, Frame -> False, Axes -> True] Out[1]= [image] ``` --- Specify an explicit origin for the axes: ```wl In[1]:= JuliaSetPlot[-0.77 + 0.22I, Frame -> False, Axes -> True, AxesOrigin -> {1.7, 1}] Out[1]= [image] ``` #### AxesStyle (4) Change the style for the axes: ```wl In[1]:= JuliaSetPlot[-0.77 + 0.22I, Frame -> False, Axes -> True, AxesStyle -> Red] Out[1]= [image] ``` --- Specify the style of each axis: ```wl In[1]:= JuliaSetPlot[-0.77 + 0.22I, Frame -> False, Axes -> True, AxesStyle -> {{Thick, Red}, {Thick, Blue}}] Out[1]= [image] ``` --- Use different styles for the ticks and the axes: ```wl In[1]:= JuliaSetPlot[-0.77 + 0.22I, Frame -> False, Axes -> True, AxesStyle -> Green, TicksStyle -> Black] Out[1]= [image] ``` --- Use different styles for the labels and the axes: ```wl In[1]:= JuliaSetPlot[-0.77 + 0.22I, Frame -> False, Axes -> True, AxesStyle -> Green, LabelStyle -> Black] Out[1]= [image] ``` #### Background (1) Use a black background as contrast with the points in the Julia set: ```wl In[1]:= JuliaSetPlot[-1, PlotStyle -> Red, Background -> Black, ColorFunction -> None] Out[1]= [image] ``` #### ColorFunction (3) Use a built-in color function to color by a scaled number of iterations: ```wl In[1]:= JuliaSetPlot[-1, ColorFunction -> Hue] Out[1]= [image] ``` Zoom in to a region to see more details: ```wl In[2]:= JuliaSetPlot[-1, ColorFunction -> Hue, PlotRange -> {{0.3, 0.45}, {0.35, 0.5}}] Out[2]= [image] ``` --- Use a named color gradient: ```wl In[1]:= JuliaSetPlot[-1, Method -> "EscapeTime", ColorFunction -> "GreenPinkTones"] Out[1]= [image] ``` --- Write a custom color function: ```wl In[1]:= JuliaSetPlot[-1, ColorFunction -> (If[#3 == 1, Purple, RGBColor[(1 - #3)^2, (1 - #3)^3, 1 - #3]]&)] Out[1]= [image] ``` #### Frame (4) ``JuliaSetPlot`` uses a frame by default: ```wl In[1]:= JuliaSetPlot[-0.125 + 0.75 I] Out[1]= [image] ``` --- Use ``Frame -> False`` to turn off the frame: ```wl In[1]:= JuliaSetPlot[-0.125 + 0.75 I, Frame -> False] Out[1]= [image] ``` --- Draw a frame on the left and right edges: ```wl In[1]:= JuliaSetPlot[-0.125 + 0.75 I, Frame -> {{True, True}, {False, False}}] Out[1]= [image] ``` --- Draw a frame on the left and bottom edges: ```wl In[1]:= JuliaSetPlot[-0.125 + 0.75 I, Frame -> {{True, False}, {True, False}}] Out[1]= [image] ``` #### FrameLabel (4) Place a label along the bottom frame of a plot: ```wl In[1]:= JuliaSetPlot[-0.125 + 0.75 I, FrameLabel -> {"label"}] Out[1]= [image] ``` --- Frame labels are placed on the bottom and left frame edges by default: ```wl In[1]:= JuliaSetPlot[-0.125 + 0.75 I, FrameLabel -> {"Bottom", "Left"}] Out[1]= [image] ``` --- Place labels on each of the edges in the frame: ```wl In[1]:= JuliaSetPlot[-0.125 + 0.75 I, FrameLabel -> {{"left", "right"}, {"bottom", "top"}}] Out[1]= [image] ``` --- Use a customized style for both labels and frame tick labels: ```wl In[1]:= JuliaSetPlot[-0.125 + 0.75 I, FrameLabel -> {{"left", "right"}, {"bottom", "top"}}, LabelStyle -> Directive[Bold, Black]] Out[1]= [image] ``` #### FrameStyle (2) Specify the style of the frame: ```wl In[1]:= JuliaSetPlot[-0.125 + 0.75 I, FrameStyle -> Directive[Black, Thick]] Out[1]= [image] ``` --- Specify the style for each frame edge: ```wl In[1]:= JuliaSetPlot[-0.125 + 0.75 I, FrameStyle -> {{Directive[Black, Thick], Directive[Red]}, {Directive[Gray, Thick], Directive[Blue]}}] Out[1]= [image] ``` #### FrameTicks (9) Frame ticks are placed automatically by default: ```wl In[1]:= JuliaSetPlot[-0.125 + 0.75 I] Out[1]= [image] ``` --- Use a frame with no ticks: ```wl In[1]:= JuliaSetPlot[-0.125 + 0.75 I, FrameTicks -> None] Out[1]= [image] ``` --- Use frame ticks on the bottom edge: ```wl In[1]:= JuliaSetPlot[-0.125 + 0.75 I, FrameTicks -> {{None, None}, {Automatic, None}}] Out[1]= [image] ``` --- By default, the top and right edges have tick marks but no tick labels: ```wl In[1]:= JuliaSetPlot[-0.125 + 0.75 I, FrameTicks -> Automatic] Out[1]= [image] ``` Use ``All`` to include tick labels on all edges: ```wl In[2]:= JuliaSetPlot[-0.125 + 0.75 I, FrameTicks -> All] Out[2]= [image] ``` --- Place tick marks at specific positions: ```wl In[1]:= JuliaSetPlot[-0.125 + 0.75 I, FrameTicks -> {{{-1, 0, 1}, Automatic}, {{-1, 1}, Automatic}}] Out[1]= [image] ``` --- Draw frame tick marks at the specified positions with specific labels: ```wl In[1]:= JuliaSetPlot[-0.125 + 0.75 I, FrameTicks -> {{{{-1, -a}, {0, 0}, {1, a}}, Automatic}, {{{-1, -a}, {1, a}}, Automatic}}] Out[1]= [image] ``` --- Specify the lengths for tick marks as a fraction of the graphics size: ```wl In[1]:= JuliaSetPlot[-0.125 + 0.75 I, FrameTicks -> {{{{-1, -a, .1}, {0, 0, .1}, {1, a, .1}}, Automatic}, {{{-1, -a, .2}, {1, a, .2}}, Automatic}}] Out[1]= [image] ``` Use different sizes in the positive and negative directions for each tick mark: ```wl In[2]:= JuliaSetPlot[-0.125 + 0.75 I, FrameTicks -> {{{{-1, -a, {.1, .1}}, {0, 0, {.1, .2}}, {1, a, {.1, .25}}}, Automatic}, {Automatic, Automatic}}] Out[2]= [image] ``` --- Specify a style for each frame tick: ```wl In[1]:= JuliaSetPlot[-0.125 + 0.75 I, FrameTicks -> {{{{-1, -a, {.1, .1}, Directive[Black, Thick, Dashed]}, {0, 0, {.1, .2}, Directive[Red]}, {1, a, {.1, .25}, Directive[Red, Thick, Dashed]}}, Automatic}, {Automatic, Automatic}}] Out[1]= [image] ``` --- Construct a function that places frame ticks at the midpoint and extremes of the frame edge: ```wl In[1]:= minMeanMax[min_, max_] := {{min, min}, {(max + min) / 2, (max + min) / 2}, {max, max}} In[2]:= JuliaSetPlot[-0.125 + 0.75 I, FrameTicks -> {{minMeanMax, None}, {minMeanMax, None}}, PlotRangePadding -> None] Out[2]= [image] ``` #### FrameTicksStyle (3) By default, the frame ticks and frame tick labels use the same styles as the frame: ```wl In[1]:= JuliaSetPlot[-0.125 + 0.75 I, FrameStyle -> Directive[Red]] Out[1]= [image] ``` --- Specify an overall style for the ticks, including the labels: ```wl In[1]:= JuliaSetPlot[-0.125 + 0.75 I, FrameTicksStyle -> Directive[Blue, Thick]] Out[1]= [image] ``` --- Use different style for the different frame edges: ```wl In[1]:= JuliaSetPlot[-0.125 + 0.75 I, FrameTicksStyle -> {{Directive[Black, Thick], Blue}, {Red, Green}}] Out[1]= [image] ``` #### ImageResolution (1) Increase ``ImageResolution`` for finer plots: ```wl In[1]:= Table[JuliaSetPlot[-1, ImageResolution -> ir], {ir, 50, 300, 50}] Out[1]= {[image], [image], [image], [image], [image], [image]} ``` #### ImageSize (7) Use named sizes such as ``Tiny``, ``Small``, ``Medium`` and ``Large`` : ```wl In[1]:= {JuliaSetPlot[z^2 - z - 1, z, ImageSize -> Tiny], JuliaSetPlot[z^2 - z - 1, z, ImageSize -> Small]} Out[1]= {[image], [image]} ``` --- Specify the width of the plot: ```wl In[1]:= {JuliaSetPlot[z^2 - z - 1, z, ImageSize -> 150], JuliaSetPlot[z^2 - z - 1, z, AspectRatio -> 1.5, ImageSize -> 150]} Out[1]= {[image], [image]} ``` Specify the height of the plot: ```wl In[2]:= {JuliaSetPlot[z^2 - z - 1, z, ImageSize -> {Automatic, 100}], JuliaSetPlot[z^2 - z - 1, z, AspectRatio -> 2, ImageSize -> {Automatic, 100}]} Out[2]= {[image], [image]} ``` --- Allow the width and height to be up to a certain size: ```wl In[1]:= {JuliaSetPlot[z^2 - z - 1, z, ImageSize -> UpTo[200]], JuliaSetPlot[z^2 - z - 1, z, AspectRatio -> 2, ImageSize -> UpTo[200]]} Out[1]= {[image], [image]} ``` --- Specify the width and height for a graphic, padding with space if necessary: ```wl In[1]:= JuliaSetPlot[z^2 - z - 1, z, ImageSize -> {200, 200}, Background -> LightBlue] Out[1]= [image] ``` Setting ``AspectRatio -> Full`` will fill the available space: ```wl In[2]:= JuliaSetPlot[z^2 - z - 1, z, AspectRatio -> Full, ImageSize -> {200, 200}, Background -> LightBlue] Out[2]= [image] ``` --- Use maximum sizes for the width and height: ```wl In[1]:= {JuliaSetPlot[z^2 - z - 1, z, ImageSize -> {UpTo[150], UpTo[100]}], JuliaSetPlot[z^2 - z - 1, z, AspectRatio -> 2, ImageSize -> {UpTo[150], UpTo[100]}]} Out[1]= {[image], [image]} ``` --- Use ``ImageSize -> Full`` to fill the available space in an object: ```wl In[1]:= Framed[Pane[JuliaSetPlot[z^2 - z - 1, z, ImageSize -> Full, Background -> LightBlue], {200, 150}]] Out[1]= [image] ``` --- Specify the image size as a fraction of the available space: ```wl In[1]:= Framed[Pane[JuliaSetPlot[z^2 - z - 1, z, ImageSize -> {Scaled[0.5], Scaled[0.5]}, Background -> LightBlue], {200, 100}]] Out[1]= [image] ``` #### PerformanceGoal (2) Generate a higher-quality plot: ```wl In[1]:= AbsoluteTiming[JuliaSetPlot[-0.125 + 0.75I, PerformanceGoal -> "Quality"]] Out[1]= {0.632358, [image]} ``` --- Emphasize performance, possibly at the cost of quality: ```wl In[1]:= AbsoluteTiming[JuliaSetPlot[-0.125 + 0.75I, PerformanceGoal -> "Speed"]] Out[1]= {0.118912, [image]} ``` #### PlotLegends (1) Legends show the number of iterations: ```wl In[1]:= JuliaSetPlot[-0.77 + 0.22I, PlotRange -> {{-0.2, 0.8}, {-0.8, 0.2}}, PlotLegends -> Automatic] Out[1]= [image] ``` #### PlotRange (2) The plot range is chosen automatically, based on the Julia set: ```wl In[1]:= JuliaSetPlot[-1] Out[1]= [image] In[2]:= JuliaSetPlot[-2 / 3] Out[2]= [image] ``` --- Zoom in to a region to see more details: ```wl In[1]:= JuliaSetPlot[-1, PlotRange -> {{0.3, 0.45}, {0.35, 0.5}}] Out[1]= [image] In[2]:= JuliaSetPlot[-1, PlotRange -> {{0.33, 0.35}, {0.44, 0.46}}] Out[2]= [image] ``` #### PlotStyle (2) By default, the points are not shown unless ``ColorFunction -> None`` : ```wl In[1]:= {JuliaSetPlot[-1], JuliaSetPlot[-1, ColorFunction -> None]} Out[1]= {[image], [image]} ``` --- Specify the plot style: ```wl In[1]:= {JuliaSetPlot[-1, PlotStyle -> Orange], JuliaSetPlot[-1, PlotStyle -> Orange, ColorFunction -> None]} Out[1]= [image] ``` #### PlotTheme (1) Use a theme with increased contrast and axes: ```wl In[1]:= JuliaSetPlot[-0.125 + 0.75I, PlotTheme -> "Scientific"] Out[1]= [image] In[2]:= JuliaSetPlot[-0.125 + 0.75I, PlotTheme -> "Scientific", ColorFunction -> None] Out[2]= [image] ``` #### Ticks (9) Ticks are placed automatically in each plot: ```wl In[1]:= JuliaSetPlot[-0.125 + 0.75 I, Frame -> False, Axes -> True] Out[1]= [image] ``` --- Use ``Ticks -> None`` to draw axes without any tick marks: ```wl In[1]:= JuliaSetPlot[-0.125 + 0.75 I, Frame -> False, Axes -> True, Ticks -> None] Out[1]= [image] ``` --- Use ticks on the $x$ axis, but not the $y$ axis: ```wl In[1]:= JuliaSetPlot[-0.125 + 0.75 I, Frame -> False, Axes -> True, Ticks -> {Automatic, None}] Out[1]= [image] ``` --- Place tick marks at specific positions: ```wl In[1]:= JuliaSetPlot[-0.125 + 0.75 I, Frame -> False, Axes -> True, Ticks -> {{-1, .6, 1}, {-1.1, 1.1}}] Out[1]= [image] ``` --- Draw tick marks at the specified positions with the specified labels: ```wl In[1]:= JuliaSetPlot[-0.125 + 0.75 I, Frame -> False, Axes -> True, Ticks -> {{{-1, -a}, {.6, b}, {1, a}}, {{-1.1, -c}, {1.1, c}}}] Out[1]= [image] ``` --- Use specific ticks on one axis and automatic ticks on the other: ```wl In[1]:= JuliaSetPlot[-0.125 + 0.75 I, Frame -> False, Axes -> True, Ticks -> {{{-1, -a}, {.6, b}, {1, a}}, Automatic}] Out[1]= [image] ``` --- Specify the lengths for ticks as a fraction of the graphics size: ```wl In[1]:= JuliaSetPlot[-0.125 + 0.75 I, Frame -> False, Axes -> True, Ticks -> {{{-1, -a, .1}, {.6, b, .1}, {1, a, .1}}, Automatic}] Out[1]= [image] ``` Use different sizes in the positive and negative directions for each tick: ```wl In[2]:= JuliaSetPlot[-0.125 + 0.75 I, Frame -> False, Axes -> True, Ticks -> {{{-1, -a, {.1, .3}}, {.6, b, {.1, .3}}, {1, a, {.1, .3}}}, Automatic}] Out[2]= [image] ``` --- Specify a style for each tick: ```wl In[1]:= JuliaSetPlot[-0.125 + 0.75 I, Frame -> False, Axes -> True, Ticks -> {{{-1, -a, .1, Red}, {.6, b, .1, Directive[Blue, Thick]}, {1, a, .1, Directive[Thick, Dashed, Red]}}, Automatic}] Out[1]= [image] ``` --- Construct a function that places ticks at the midpoint and extremes of the axis: ```wl In[1]:= minMeanMax[min_, max_] := {{min, min}, {(max + min) / 2, (max + min) / 2}, {max, max}} In[2]:= JuliaSetPlot[-0.125 + 0.75 I, Frame -> False, Axes -> True, Ticks -> {minMeanMax, minMeanMax}, PlotRangePadding -> None] Out[2]= [image] ``` #### TicksStyle (4) By default, the ticks and tick labels use the same style as the axis: ```wl In[1]:= JuliaSetPlot[-0.125 + 0.75 I, Frame -> False, Axes -> True, AxesStyle -> Red] Out[1]= [image] ``` --- Specify an overall tick style, including the tick labels: ```wl In[1]:= JuliaSetPlot[-0.125 + 0.75 I, Frame -> False, Axes -> True, TicksStyle -> Red] Out[1]= [image] ``` --- Specify the tick style for each of the axes: ```wl In[1]:= JuliaSetPlot[-0.125 + 0.75 I, Frame -> False, Axes -> True, TicksStyle -> {Directive[White, Thick], Directive[Red, Bold]}] Out[1]= [image] ``` --- Use a different style for the tick labels and tick marks: ```wl In[1]:= JuliaSetPlot[-0.125 + 0.75 I, Frame -> False, Axes -> True, TicksStyle -> Directive[Red, Thick], LabelStyle -> White] Out[1]= [image] ``` ### Possible Issues (4) ``JuliaSetPlot`` only works with rational functions: ```wl In[1]:= JuliaSetPlot[Sin[z], z] ``` JuliaSetPlot::polyesctime: Escape time method works only for polynomials ```wl Out[1]= JuliaSetPlot[Sin[z], z] ``` --- When using orbit detection, sometimes no attractive orbits are detected: ```wl In[1]:= JuliaSetPlot[z / (z ^ 2 - 1), z, Method -> "OrbitDetection"] ``` JuliaSetPlot::polyesctime: Escape time method works only for polynomials JuliaSetPlot::noorbs: No attractive behavior detected ```wl Out[1]= JuliaSetPlot[(z/-1 + z^2), z, Method -> "OrbitDetection"] ``` --- If the value of the ``"Bound"`` method option is too low for a rational function, no points may be returned: ```wl In[1]:= JuliaSetPlot[(500 / z ^ 2 - 1), z] Out[1]= [image] In[2]:= JuliaSetPlot[(500 / z ^ 2 - 1), z, Method -> {"Bound" -> 9}, Axes -> True, PlotStyle -> Automatic, ColorFunction -> None] Out[2]= [image] ``` --- Some very large Julia sets can take a long time to compute: ```wl In[1]:= AbsoluteTiming[JuliaSetPlot[(z ^ 2 - 5 / 19/z ^ 2 + 5 / 19), z, PlotStyle -> Automatic, ColorFunction -> None]] Out[1]= [image] ``` ### Neat Examples (1) Colorize the components of a Julia set: ```wl In[1]:= Colorize[MorphologicalComponents[JuliaSetPlot[z ^ 2 + (-0.123 + 0.745I), z, PlotStyle -> Automatic, ColorFunction -> None, Frame -> False, Method -> {"ClosenessTolerance" -> 0.001}], .5]] Out[1]= [image] ``` ## See Also * [`MandelbrotSetPlot`](https://reference.wolfram.com/language/ref/MandelbrotSetPlot.en.md) * [`JuliaSetIterationCount`](https://reference.wolfram.com/language/ref/JuliaSetIterationCount.en.md) * [`ListPlot`](https://reference.wolfram.com/language/ref/ListPlot.en.md) * [`ArrayPlot`](https://reference.wolfram.com/language/ref/ArrayPlot.en.md) * [`DensityPlot`](https://reference.wolfram.com/language/ref/DensityPlot.en.md) * [`ContourPlot`](https://reference.wolfram.com/language/ref/ContourPlot.en.md) ## Related Guides * [Iterated Maps & Fractals](https://reference.wolfram.com/language/guide/IteratedMapsAndFractals.en.md) * [Complex Visualization](https://reference.wolfram.com/language/guide/ComplexVisualization.en.md) ## History * [Introduced in 2014 (10.0)](https://reference.wolfram.com/language/guide/SummaryOfNewFeaturesIn100.en.md)