Mickey Mouse's Mandelbrot Set

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Mickey Mouse's Mandelbrot set Sarah Boyt George Mason University Math 401 Mathematics and 3D Printing Brief History: Once upon a time, in the early 1800s mathematician Pierre Fatou became interested in the iteration of the equation c→c^2+k where c and k are complex numbers and k a constant. Later Gaston Julia studied how iterating this formula created a set with a boundary of infinite length that was impossible to draw by hand (even though it is a finite curve).This set is now known as the Julia Set. In the late 1970s and early 1980s, Benoit Mandelbrot expanded on the work of Fatou and Julia. Mandelbrot used a computer to plot Julia Sets. While investigating the Julia Set, Mandelbrot studied the z→z^2+c formula and publicized the set which is now known as the Mandelbrot Set. In the 1980s mathematicians and programmers became interested in turning the Mandelbrot Set into 3 dimensions. However, there were several challenges.Computers in the 1980s were unable to handle the calculations. Additionally, there is no 3D equivalent to the 2D complex product (a+bi)·(c+di). In 2007, a controversial algorithm using algebra based on spherical coordinates triplets {ρ, φ, θ} (module, longitude and latitude) was discovered to produce beautiful graphics of the Mandelbrot Set . Although the process is not correct from a mathematical perspective the images became iconic. The 3D version, especially when raised to higher polynomials z→z^n+c is known as the Mandelbulb. Mandelbrot and Julia Set Connection: The Mandelbrot and Julia Set both use the recursive formula Z=Z(n-1)+c. For the Mandelbrot Set z0=0. c is any number on the complex plane. If Zn ,where n is the nth iteration, diverges then c is not part of the Mandelbrot Set. The color of the point c in a graphic depends on how fast Z grows. If Zn is bounded (meaning it does not exceed absolute value 2 or 2i), then c is part of the Mandelbrot Set. In most graphics, if c is part of the Mandelbrot Set, c is colored black. The Julia Set is very similar. However, c is fixed and z0 varies. As with the Mandelbrot Set, if Zn is bounded (meaning it does not exceed absolute value 2 or 2i), then c is part of the Julia Set. Otherwise c is not part of the Julia Set. Like with the Mandelbrot Set, the points are color coded in a graphic. If you pick a c in the Mandelbrot Set, then you can use it to generate a Julia Set. My Mandelbrot Set: For my Mandelbrot Set, I took the base code provided by my professor and altered the power of z. After playing around with different values, I realized that I didn't want to do an integer power because it was pretty but predictable. I ended up going with Z=Z^4.5+c because I liked that the final outcome was symmetrical but not predictable. Note: Before printing the piece will need to be scaled. All you need to do is scale x to the desired size and y,z will scale accordingly. References: https://archive.bridgesmathart.org/2010/bridges2010-247.pdf https://www.youtube.com/watch?v=mg4bp7G0D3s&t=217s https://en.wikipedia.org/wiki/Benoit_Mandelbrot

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