MandelbrotSetPlot[{zmin,zmax}]
plots the portion of the Mandelbrot set inside the rectangle with corners zmin and zmax.
plots the Mandelbrot set over a default rectangle.
MandelbrotSetPlot
MandelbrotSetPlot[{zmin,zmax}]
plots the portion of the Mandelbrot set inside the rectangle with corners zmin and zmax.
plots the Mandelbrot set over a default rectangle.
Details and Options
- The Mandelbrot set is the set of all complex numbers c for which the sequence
does not diverge to infinity when starting with
. - The default rectangle for MandelbrotSetPlot[] has corners
and
. - MandelbrotSetPlot produces a Graphics object containing a Raster primitive.
- MandelbrotSetPlot[] has the same options as Graphics, with the following additions:
-
ColorFunction Automatic how to determine the color of a pixel EscapeRadius 2 how to determine that a point is not in the set Frame True whether to draw a frame around the plot ImageResolution 500 resolution of the image in the larger direction MaxIterations 1000 maximum number of iterates per point PerformanceGoal $PerformanceGoal aspects of performance to try to optimize PlotLegends None legends for the number of interactions PlotTheme $PlotTheme overall theme for the plot - With MaxIterations->n, where n is a positive integer, the function
will be iterated at most n times to determine if the orbit of 0 ever exceeds 2. - 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"].
Examples
open all close allBasic Examples (3)
Show the whole Mandelbrot set:
MandelbrotSetPlot[]MandelbrotSetPlot[{-0.65 + 0.47I, -0.4 + 0.72I}, MaxIterations -> 200, ColorFunction -> "RedBlueTones"]Show a legend of the number of iterations:
MandelbrotSetPlot[{-0.65 + 0.47I, -0.4 + 0.72I}, PlotLegends -> Automatic]Options (59)
AspectRatio (4)
By default, MandelbrotSetPlot uses the same width and height:
MandelbrotSetPlot[]Use numerical value to specify the height-to-width ratio:
MandelbrotSetPlot[AspectRatio -> .5]AspectRatioAutomatic determines the ratio from the plot ranges:
MandelbrotSetPlot[AspectRatio -> Automatic]AspectRatioFull adjusts the height and width to tightly fit inside other constructs:
plot = MandelbrotSetPlot[AspectRatio -> Full];
{Framed[Pane[plot, {75, 100}]], Framed[Pane[plot, {100, 100}]], Framed[Pane[plot, {100, 75}]]}Axes (3)
By default, MandelbrotSetPlot uses a frame instead of axes:
MandelbrotSetPlot[]MandelbrotSetPlot[Frame -> False, Axes -> True]Turn each axis on individually:
{MandelbrotSetPlot[Frame -> False, Axes -> {True, False}], MandelbrotSetPlot[Frame -> False, Axes -> {False, True}]}AxesLabel (3)
AxesOrigin (2)
AxesStyle (4)
Change the style for the axes:
MandelbrotSetPlot[Frame -> False, Axes -> True, AxesStyle -> Red]Specify the style of each axis:
MandelbrotSetPlot[Frame -> False, Axes -> True, AxesStyle -> {{Thick, Orange}, {Thick, Blue}}]Use different styles for the ticks and the axes:
MandelbrotSetPlot[Frame -> False, Axes -> True, AxesStyle -> Blue, TicksStyle -> Red]Use different styles for the labels and the axes:
MandelbrotSetPlot[Frame -> False, Axes -> True, AxesStyle -> Blue, LabelStyle -> Red]ColorFunction (5)
Use a built-in color function to color by a scaled number of iterates:
MandelbrotSetPlot[ColorFunction -> Hue]Modify a built-in color function:
MandelbrotSetPlot[ColorFunction -> (If[#3 == 1, Blue, GrayLevel[5#3]]&)]MandelbrotSetPlot[ColorFunction -> "GreenPinkTones"]Zoom in to see more detail and colors:
MandelbrotSetPlot[{0.2 + 0.45I, 0.4 + 0.65I}, ColorFunction -> "GreenPinkTones"]Write a custom color function:
MandelbrotSetPlot[ColorFunction -> (If[#1 == 1, Black, RGBColor[(1 - #3)^2, (1 - #3)^3, 1 - #3]]&)]Color all points that escape in less than 10 iterations:
MandelbrotSetPlot[MaxIterations -> 10, ColorFunction -> (If[#3 > 9 / 10, Black, Blue]&)]ColorFunctionScaling (2)
EscapeRadius (1)
Frame (3)
MandelbrotSetPlot[Frame -> False]Draw a frame on the left and right edges:
MandelbrotSetPlot[Frame -> {{True, True}, {False, False}}]Draw a frame on the left and bottom edges:
MandelbrotSetPlot[Frame -> {{True, False}, {True, False}}]FrameLabel (4)
Place a label along the bottom frame of a plot:
MandelbrotSetPlot[FrameLabel -> {"Real"}]MandelbrotSetPlot[FrameLabel -> {"Real", "Imaginary"}]Place labels on each of the edges in the frame:
MandelbrotSetPlot[FrameLabel -> {{"left", "right"}, {"bottom", "top"}}]Use a customized style for both labels and frame tick labels:
MandelbrotSetPlot[FrameLabel -> {{y, y2}, {x, x2}}, LabelStyle -> Directive[Bold, StandardGray]]FrameStyle (2)
FrameTicks (8)
MandelbrotSetPlot[FrameTicks -> None]Use frame ticks on the bottom edge:
MandelbrotSetPlot[FrameTicks -> {{None, None}, {Automatic, None}}]By default, the top and right edges have tick marks but no tick labels:
MandelbrotSetPlot[FrameTicks -> Automatic]Use All to include tick labels on all edges:
MandelbrotSetPlot[FrameTicks -> All]Place tick marks at specific positions:
MandelbrotSetPlot[FrameTicks -> {{{.1, .5, 1}, {.1, .5, 1}}, {{-.5, 0, .5}, {-.5, 0, .5}}}]Draw frame tick marks at specified positions with specific labels:
MandelbrotSetPlot[FrameTicks -> {{{{.1, a}, {.5, b}, {1, c}}, {.1, .5, 1}}, {{{-.5, c}, {0, d}, {.5, e} }, {-.5, 0, .5 }}}]Specify the lengths for tick marks as a fraction of the graphics size:
MandelbrotSetPlot[FrameTicks -> {{Automatic, {{.1, a, .12}, {.5, b, .1}, {1, c, .25}}}, {Automatic, Automatic}}]Use different sizes in the positive and negative directions for each tick mark:
MandelbrotSetPlot[Axes -> False, FrameTicks -> {{Automatic, {{.1, a, {.12, .05}}, {.5, b, {.1, .1}}, {1, c, {.25, .2}}}}, {Automatic, Automatic}}]Specify a style for each frame tick:
MandelbrotSetPlot[FrameTicks -> {{Automatic, {{.1, a, {.12, .05}, Directive[Blue, Thick, Dashed]}, {.5, b, {.1, .1}}, {1, c, {.25, .2}, Directive[Thick, Green]}}}, {Automatic, Automatic}}]Construct a function that places frame ticks at the midpoint and extremes of the frame edge:
minMeanMax[min_, max_] := {{min, min}, {(max + min) / 2, (max + min) / 2}, {max, max}}MandelbrotSetPlot[FrameTicks -> {{minMeanMax, None}, {minMeanMax, None}}, PlotRangePadding -> None]ImageResolution (1)
Increase ImageResolution for finer plots:
Table[MandelbrotSetPlot[ImageResolution -> ir], {ir, 50, 300, 50}]ImageSize (7)
Use named sizes such as Tiny, Small, Medium and Large:
{MandelbrotSetPlot[ImageSize -> Tiny], MandelbrotSetPlot[ImageSize -> Small]}Specify the width of the plot:
{MandelbrotSetPlot[ImageSize -> 150], MandelbrotSetPlot[AspectRatio -> 1.5, ImageSize -> 150]}Specify the height of the plot:
{MandelbrotSetPlot[ImageSize -> {Automatic, 150}], MandelbrotSetPlot[AspectRatio -> 2, ImageSize -> {Automatic, 150}]}Allow the width and height to be up to a certain size:
{MandelbrotSetPlot[ImageSize -> UpTo[200]], MandelbrotSetPlot[AspectRatio -> 2, ImageSize -> UpTo[200]]}Specify the width and height for a graphic, padding with space if necessary:
MandelbrotSetPlot[ImageSize -> {200, 250}, Background -> Lighter@Orange]Setting AspectRatioFull will fill the available space:
MandelbrotSetPlot[AspectRatio -> Full, ImageSize -> {200, 250}, Background -> Lighter@Orange]Use maximum sizes for the width and height:
{MandelbrotSetPlot[ImageSize -> {UpTo[150], UpTo[100]}], MandelbrotSetPlot[AspectRatio -> 2, ImageSize -> {UpTo[150], UpTo[100]}]}Use ImageSizeFull to fill the available space in an object:
Framed[Pane[MandelbrotSetPlot[ImageSize -> Full, Background -> Lighter@Orange], {200, 200}]]Specify the image size as a fraction of the available space:
Framed[Pane[MandelbrotSetPlot[AspectRatio -> Full, ImageSize -> {Scaled[0.5], Scaled[0.5]}, Background -> Lighter@Orange], {200, 100}]]MaxIterations (1)
Increase MaxIterations to improve quality when zooming in:
{MandelbrotSetPlot[{-0.743 + 0.175I, -0.723 + 0.195I}], MandelbrotSetPlot[{-0.743 + 0.175I, -0.723 + 0.195I},
MaxIterations -> 800]}PlotTheme (1)
Ticks (4)
Ticks are placed automatically in each plot:
MandelbrotSetPlot[{.25, .5 + .25 I}, MaxIterations -> 50, Frame -> False, Axes -> True]Use TicksNone to not draw any tick marks:
MandelbrotSetPlot[{.25, .5 + .25 I}, MaxIterations -> 50, Frame -> False, Axes -> True, Ticks -> None]Place tick marks at specific positions:
MandelbrotSetPlot[{.25, .5 + .25 I}, MaxIterations -> 50, Frame -> False, Axes -> True, Ticks -> {{.3, .4, .45}, {.05, .1, .2}}]Draw tick marks at the specified positions with the specified labels:
MandelbrotSetPlot[{.25, .5 + .25 I}, MaxIterations -> 50, Frame -> False, Axes -> True, Ticks -> {{{.3, a}, {.4, b}, {.45, c}}, {{.05, d}, {.1, e}, {.2, f}}}]TicksStyle (4)
Specify overall ticks style, including the tick labels:
MandelbrotSetPlot[{.25, .5 + .25 I}, MaxIterations -> 50, Frame -> False, Axes -> True, TicksStyle -> Directive[Blue, 14]]Specify tick style for each of the axes:
MandelbrotSetPlot[{.25, .5 + .25 I}, MaxIterations -> 50, Frame -> False, Axes -> True, TicksStyle -> {Directive[Red, 15], Directive[Blue, 15]}]Specify tick marks with scaled lengths:
MandelbrotSetPlot[{.25, .5 + .25 I}, MaxIterations -> 50, Frame -> False, Axes -> True, TicksStyle -> Directive[Darker@Green, 15], Ticks -> {{{.3, a, .1}, {.4, b, .05}, {.45, c, .2}}, {{.05, d, .3}, {.1, e, .4}, {.2, f, .5}}}]Customize each tick with position, length, labeling and styling:
MandelbrotSetPlot[{.25, .5 + .25 I}, MaxIterations -> 50, Frame -> False, Axes -> True, Ticks -> {{{.3, a, .1, Directive[Thick, White]}, {.4, b, .35, Directive[Thick, White]}, {.45, c, .7, Directive[Thick, White]}}, {{.05, d, .3, Directive[Thick, White]}, {.1, e, .4, Directive[Thick, White]}, {.2, f, .5, Directive[Thick, White]}}}]Properties & Relations (3)
Points in the Mandelbrot set determine quadratic Julia sets:
c = -0.123 + 0.745I;
{MandelbrotSetPlot[Epilog -> {Red, PointSize[Medium], Point[{Re[c], Im[c]}]}],
JuliaSetPlot[c]}Use MandelbrotSetMemberQ to determine whether a point is in the Mandelbrot set:
MandelbrotSetPlot[Epilog -> {Red, PointSize[Medium], Point[{-0.123, 0.745}]}]MandelbrotSetMemberQ[-0.123 + 0.745I]MandelbrotSetIterationCount gives the number of iterations used to determine if a point is not in the Mandelbrot set:
MandelbrotSetPlot[Epilog -> {Red, PointSize[Medium], Point[{-0.123, 0.945}]}]MandelbrotSetIterationCount[-0.123 + 0.945I, MaxIterations -> 100]See Also
JuliaSetPlot MandelbrotSetIterationCount MandelbrotSetMemberQ MandelbrotSetBoettcher ArrayPlot DensityPlot ContourPlot
Function Repository: MandelbrotSetRemap
Related Guides
History
Text
Wolfram Research (2014), MandelbrotSetPlot, Wolfram Language function, https://reference.wolfram.com/language/ref/MandelbrotSetPlot.html.
CMS
Wolfram Language. 2014. "MandelbrotSetPlot." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/MandelbrotSetPlot.html.
APA
Wolfram Language. (2014). MandelbrotSetPlot. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/MandelbrotSetPlot.html
BibTeX
@misc{reference.wolfram_2026_mandelbrotsetplot, author="Wolfram Research", title="{MandelbrotSetPlot}", year="2014", howpublished="\url{https://reference.wolfram.com/language/ref/MandelbrotSetPlot.html}", note=[Accessed: 13-June-2026]}
BibLaTeX
@online{reference.wolfram_2026_mandelbrotsetplot, organization={Wolfram Research}, title={MandelbrotSetPlot}, year={2014}, url={https://reference.wolfram.com/language/ref/MandelbrotSetPlot.html}, note=[Accessed: 13-June-2026]}