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MandelbrotSetDistance[c]

estimates the distance from c to the nearest point in the Mandelbrot set.

MandelbrotSetDistance[c,"Interior"]

estimates the distance from c to the nearest point in the complement of the Mandelbrot set.

Details and Options
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Options  
MaxIterations  
WorkingPrecision  
Properties & Relations  
Neat Examples  
See Also
Related Guides
History
Cite this Page

MandelbrotSetDistance

MandelbrotSetDistance[c]

estimates the distance from c to the nearest point in the Mandelbrot set.

MandelbrotSetDistance[c,"Interior"]

estimates the distance from c to the nearest point in the complement of the Mandelbrot set.

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 .
  • MandelbrotSetDistance takes the following options:
  • MaxIterations 100maximum number of iterations to use
    WorkingPrecision Automaticthe precision used in internal computations
  • With the option MaxIterations->n, the sequence will be iterated at most n times to determine the distance. The default setting is MaxIterations->100.
  • As MaxIterations approaches infinity, MandelbrotSetDistance[c] will converge on a number that is guaranteed to be at least the true distance.

Examples

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Basic Examples  (2)

Estimate the distance from to the Mandelbrot set:

Wolfram Language code: MandelbrotSetDistance[1 + I]

MandelbrotSetDistance is useful for making stark binary images of the Mandelbrot set:

Wolfram Language code: data = Table[MandelbrotSetDistance[x + I * y], {y, -1.5, 1.5, .005}, {x, -2, 1, .005}];
Wolfram Language code: ArrayPlot[data /. {_ ? (0 < # < 0.008&) -> 1, _ ? (# > 0.008&) -> 0}]

Scope  (2)

Estimate the distance from to the Mandelbrot set:

Wolfram Language code: MandelbrotSetDistance[.73 - .9 * I]

The origin is in the Mandelbrot set so the distance is zero:

Wolfram Language code: MandelbrotSetDistance[0]

Estimate the distance from the origin to the boundary of the Mandelbrot set:

Wolfram Language code: MandelbrotSetDistance[0, "Interior"]

Options  (2)

MaxIterations  (1)

Sometimes MaxIterations needs to be increased to get a nonzero distance:

Wolfram Language code: {MandelbrotSetDistance[25745 / 100000, MaxIterations -> 30], MandelbrotSetDistance[25745 / 100000, MaxIterations -> 50000]}

WorkingPrecision  (1)

Increase WorkingPrecision to get a more precise answer:

Wolfram Language code: MandelbrotSetDistance[9 / 7]
Wolfram Language code: MandelbrotSetDistance[9 / 7, WorkingPrecision -> 50]

Properties & Relations  (1)

Wolfram Language code: MandelbrotSetDistance[100]
Wolfram Language code: MandelbrotSetDistance[101]

Neat Examples  (4)

Combine interior and exterior distance information:

Wolfram Language code: data = Table[Max[MandelbrotSetDistance[#, "Interior"], MandelbrotSetDistance[#] / 4]&[x + I * y], {y, -1, 1, .02}, {x, -2, .6, .02}];
Wolfram Language code: ListPlot3D[data]

Display the distance to the Mandelbrot set from points on one-quarter of the unit circle:

Wolfram Language code: data = Table[MandelbrotSetDistance[E ^ (I t Pi), MaxIterations -> 120], {t, 0, Pi / 2, .001}];
Wolfram Language code: ListPlot[data]

Represent the distance to the complement of the Mandelbrot set as height:

Wolfram Language code: data = Table[MandelbrotSetDistance[x + I * y, "Interior", MaxIterations -> 100], {x, -2., 1., .031}, {y, -1.5, 1.5, .031}];
Wolfram Language code: ListPlot3D[data, Mesh -> None, PlotRange -> Full]

Display distance to the Mandelbrot set as height:

Wolfram Language code: data = Table[MandelbrotSetDistance[x + I * y, MaxIterations -> 120], {y, -1.5, 1.5, .01}, {x, -2, 1, .01}];
Wolfram Language code: ListPlot3D[data, Mesh -> None]

Display the distance from the exterior of the Mandelbrot set using color:

Wolfram Language code: Graphics@Raster[data /. {_ ? (# <= 0.&) -> .9}, ColorFunction -> Hue]
Wolfram Research (2014), MandelbrotSetDistance, Wolfram Language function, https://reference.wolfram.com/language/ref/MandelbrotSetDistance.html.

Text

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

CMS

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

APA

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

BibTeX

@misc{reference.wolfram_2026_mandelbrotsetdistance, author="Wolfram Research", title="{MandelbrotSetDistance}", year="2014", howpublished="\url{https://reference.wolfram.com/language/ref/MandelbrotSetDistance.html}", note=[Accessed: 13-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_mandelbrotsetdistance, organization={Wolfram Research}, title={MandelbrotSetDistance}, year={2014}, url={https://reference.wolfram.com/language/ref/MandelbrotSetDistance.html}, note=[Accessed: 13-June-2026]}