Heart Shaped Mandelbrot Set

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Heart Shaped Mandelbrot Set Marlanna Bozicevich 10/25/2021 George Mason University Math401: Mathematics Through 3D Printing Background The idea of the Mandelbrot set was first introduced within the origin of complex dynamics by French mathematicians Pierre Fatou and Gaston Julia in the early 1900s. The first rendition and drawing were made by Robert Brooks and Peter Matelski in 1978. Then, in 1980, at IBM’s Thomas J Watson Research Center, Benoit Mandelbrot created the first computer graphic with full visualization of the Mandelbrot set. He was one of the first to use technology to display visual complexity through very simplistic rules. In general, the Mandelbrot Set illustrates a set of points in the complex plane where the corresponding Julia set is connected and not computable. This set of points is obtained from the following equation: z_n+1 = (z_n)^2 + C where C represents the complex numbers for which the orbit of zn does not diverge to infinity and z_0 = C. By applying the iteration repeatedly, the absolute value of zn remains unbounded for all n > 0. Process When examining the code for the Mandelbrot set, I noticed that as you increase z’s exponent, the overall shape of the Mandelbrot set turns very circular and symmetrical. I thought this was too predictable and somewhat boring. I then decided to use decimal exponents instead and began to find more and more unique, yet still symmetrical shapes. After a long process of testing various exponential values, I decided on using z^2.25 because it produced this unique heart shape from a Birdseye view with added circles that were still symmetric. I have included an image of my Mathematica code. However, when running this code, be aware that in order to generate the 3D plot and export the model as an STL file, the run time takes about 10 minutes. Printing This three-dimensional print took about 6 hours and 15 minutes on the Makerbot Mini with PLA filament. I did however run into a few issues at the start of the print. First, scaling had to be done to change the sizing of my object. I changed the x and y direction to be 7% of the original size, and z direction to be 15% of the original. This made the dimensions about 70x70x15 mm. Additionally, the automatic setting for printer temperature was 220 even though PLA filament melts around 225. Unfortunately, we did not catch this issue until the print had started. We had to restart the printer entirely and adjust the settings in a new file with a printer temperature of 215. Additionally, we made a change in print speed from 40 to 30 due to the large number of intricate details. No supports, nor rafts were used for this print. Citations https://mathworld.wolfram.com/MandelbrotSet.html https://en.wikipedia.org/wiki/Benoit_Mandelbrot https://en.wikipedia.org/wiki/Mandelbrot_set#History https://math.bu.edu/DYSYS/FRACGEOM/node2.html http://pi.math.cornell.edu/~lipa/mec/lesson5.html