Mandelbrot Set (2017)

A NetLogo ( Citation: Wilensky, 1999 Wilensky, U.(1999). Retrieved from http://ccl.northwestern.edu/netlogo/ ) model by Rik Blok.

Explore the Mandelbrot Set and these related fractals:

• Multibrot set
• Mandelbar set
• Burning ship

1 Demonstration

This movie was made by turning on the record-png switch and the zoom-every button, and setting the jump-size to 1%. The sequence of PNG images were stitched together and set to Rossini’s William Tell Overture Finale with Windows Live Movie Maker.

2 Mappings

Each fractal is defined by a function in the complex plane. Starting with an initial value z=0, each point c in the plane is repeatedly iterated through the map, z → f(z,c). The mapping function is characterized by an exponent, d (d-multibrot-exp slider in the simulation), as follows:

• Mandelbrot: f(z,c) = z² + c (same as Multibrot with d=2)
• Multibrot: f(z,c) = z^d + c
• Mandelbar: f(z,c) = Conj(z)^d + c
• Burning ship: f(z,c) = (|Re(z)| + i |Im(z)|)^d + c

A point c is excluded from the set if the value z diverges after repeated iteration. In the simulation, excluded points are painted a color indicating how many iterations were required to decide they have diverged. Black points indicate undecided candidates that may belong to the set.

3 Other implementations and examples

This implementation of the Mandelbrot set is neither fast nor beautiful – it’s just a proof of concept and a demonstration of how to code in NetLogo. If you’re interested in the Mandelbrot set or similar fractals, check out these excellent pages:

• Google’s Julia Map
• Last Lights On - video of Mandelbrot zoom to 10²²⁸

4 References