# MandelbrotSetDistance

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 .
• With the option , the sequence will be iterated at most n times to determine the distance. The default setting is MaxIterations->100.
• As MaxIterations approaches infinity, will converge on a number that is guaranteed to be at least the true distance.

# Examples

open allclose all

## Basic Examples(2)

Estimate the distance from to the Mandelbrot set:

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

## Scope(2)

Estimate the distance from to the Mandelbrot set:

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

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

## Options(2)

### MaxIterations(1)

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

### WorkingPrecision(1)

Increase WorkingPrecision to get a more precise answer:

## Neat Examples(4)

Combine interior and exterior distance information:

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

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

Display distance to the Mandelbrot set as height:

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

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_2022_mandelbrotsetdistance, author="Wolfram Research", title="{MandelbrotSetDistance}", year="2014", howpublished="\url{https://reference.wolfram.com/language/ref/MandelbrotSetDistance.html}", note=[Accessed: 09-June-2023 ]}

#### BibLaTeX

@online{reference.wolfram_2022_mandelbrotsetdistance, organization={Wolfram Research}, title={MandelbrotSetDistance}, year={2014}, url={https://reference.wolfram.com/language/ref/MandelbrotSetDistance.html}, note=[Accessed: 09-June-2023 ]}