MandelbrotSetIterationCount

MandelbrotSetIterationCount[c]

returns the number of iterations of the function , beginning with , that are needed to determine whether c is in 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 .
  • With the option MaxIterations->n, the sequence will be iterated at most n times to determine if the sequence diverges.
  • The default setting is MaxIterations->1000.
  • If the maximum number of iterations is reached, z is assumed to be in the Mandelbrot set.
  • For some points that the algorithm knows in advance to be inside the Mandelbrot set, MandelbrotSetIterationCount will return one greater than the value of MaxIterations.

Examples

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Basic Examples  (4)

Find the number of iterations needed to determine that is not in the Mandelbrot set:

Zero is known to be inside the Mandelbrot set and therefore returns :

It takes a few additional iterations to determine that 0.250001 is not in the Mandelbrot set:

MandelbrotSetIterationCount works on all kinds of numbers:

Options  (2)

MaxIterations  (1)

Sometimes MaxIterations needs to be increased to get the true number of iterations:

WorkingPrecision  (1)

Sometimes increasing WorkingPrecision gives a more accurate result:

Properties & Relations  (1)

ArrayPlot applied to MandelbrotSetIterationCount is essentially MandelbrotSetPlot:

Possible Issues  (1)

With MaxIterations->Infinity, the calculation may not converge in a finite number of steps:

Since is known to be in the Mandelbrot set, it returns DirectedInfinity:

Neat Examples  (3)

Make a Histogram of iteration counts along the imaginary axis:

Display the iteration count as height:

Half of a rotated Mandelbrot set:

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

Text

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

CMS

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

APA

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

BibTeX

@misc{reference.wolfram_2024_mandelbrotsetiterationcount, author="Wolfram Research", title="{MandelbrotSetIterationCount}", year="2014", howpublished="\url{https://reference.wolfram.com/language/ref/MandelbrotSetIterationCount.html}", note=[Accessed: 22-December-2024 ]}

BibLaTeX

@online{reference.wolfram_2024_mandelbrotsetiterationcount, organization={Wolfram Research}, title={MandelbrotSetIterationCount}, year={2014}, url={https://reference.wolfram.com/language/ref/MandelbrotSetIterationCount.html}, note=[Accessed: 22-December-2024 ]}