# MandelbrotSetMemberQ

returns True if z is in the Mandelbrot set, and False otherwise.

# Details and Options

• The Mandelbrot set is the set of all complex numbers for which the sequence does not diverge to infinity when starting with .
• With the option , 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.

# Examples

open allclose all

## Basic Examples(3)

Test whether is a member of the Mandelbrot set:

Zero is known to be inside the Mandelbrot set:

It takes a few hundred iterations to determine that 0.2501 is not in the Mandelbrot set:

## Scope(2)

MandelbrotSetMemberQ threads itself element-wise over lists:

MandelbrotSetMemberQ works on all kinds of numbers:

## Options(1)

### MaxIterations(1)

Sometimes MaxIterations needs to be increased to eliminate false positives:

## Applications(2)

Generate the Mandelbrot set from randomly chosen points:

Approximate the first point along the line that is not in the Mandelbrot set:

Show the point on the Mandelbrot set:

## Possible Issues(1)

With , the calculation may not converge in a finite number of steps:

## Neat Examples(4)

MandelbrotSetMemberQ can be used to get an estimate of the area of the Mandelbrot set:

Display the Julia sets for points in the Mandelbrot set:

Rotate the Mandelbrot set:

Use MandelbrotSetMemberQ to distinguish Julia sets that are Cantor sets:

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

#### Text

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

#### CMS

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

#### APA

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

#### BibTeX

@misc{reference.wolfram_2021_mandelbrotsetmemberq, author="Wolfram Research", title="{MandelbrotSetMemberQ}", year="2014", howpublished="\url{https://reference.wolfram.com/language/ref/MandelbrotSetMemberQ.html}", note=[Accessed: 24-January-2022 ]}

#### BibLaTeX

@online{reference.wolfram_2021_mandelbrotsetmemberq, organization={Wolfram Research}, title={MandelbrotSetMemberQ}, year={2014}, url={https://reference.wolfram.com/language/ref/MandelbrotSetMemberQ.html}, note=[Accessed: 24-January-2022 ]}