The Julia Set: Storm Cloud Edition

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David Lampton 9/10/2021 George Mason University Math 401 Mathematics Through 3D Printing Assignment 5 Hey everyone, here is my second print! It is a model of part of the Julia Set. The Julia Set is the closure of the set of repelling periodic points of a polynomial P with a domain and range of complex values. A periodic point is a point in a function that the function returns to after a certain number of iterations, and it is repelling if the absolute value of the product of P raised to a power and it, is greater than one. Another interesting tidbit about the Julia set is the set’s relationship with the point (0,0). 0 is the only point in the set that lacks a succession of preimages that converge to the Julia Set, so the Julia Set acts as a chaotic repeller for the point 0. In other words, J(P) is not a smooth curve as in the case of Q (0). Interestingly, when one types in 0 in the Javascript Julia Set Generator, one is met with a perfect circle. Perhaps there is some correspondence here. The website’s description also notes that the Mandelbrot Set is a dictionary of all quadratic Julia Sets. Similar to the Julia Set, the Mandelbrot Set is the set of complex numbers that do not diverge to infinity when iterated using the formula: fc(z)=z2+c. Since the model of the Mandelbrot Set is two-dimensional, visualizing it in three dimensions is challenging. Mathematicians Paul Nylander and Daniel White have used formulas involving Cartesian and Spherical coordinate systems to best create a 3D model of the Mandelbrot Set. In a technical sense, their model cannot be considered a 3D version of the Mandelbrot Set, but the model looks impressive and has been received positively, nonetheless. On another note, the study of 3D Mandelbrot Set modeling gave rise to a breathtaking, new type of 3D fractal named the Mandelbulb, a courtesy mainly by White and Nylander. I want to also make it known that I researched the Mandelbrot/Julia Sets in English 302 and watched YouTube videos about them before this class, so I am glad I get to see them again. When it came time to code, I initially wanted to model a higher dimension of the Mandelbrot Set and make something that looked like a snowball. However, I had some difficulty getting the code to work and render an image, so I decided to shift gears to the Julia Set instead. After not producing a cool shape with higher dimension Julia Sets (I tried z^3, z^6 and random c values etc. but all I got were circular or square mounds), I decided to stick with z=z^2+c. The range of x and y values for my plot are from -2 to 2, and the plot is of the Julia Set function z=z^2+c with c=-0.18-0.48I over 30 iterations. I produced the values in c when I requested Wolfram Alpha give two random integers between -100 and 0. It gave me -18 and then -48, so I used those values in the c equation in that order. Finally, it is worth noting that any equations/codes I used were either built-in Mathematica codes or those provided by my instructor. In the printing stage, I used the MakerBot printer, and the print took around 2 hours to complete. I doubt I used any raft or supports other than the base that came with my print. Even though I wanted to have a blue snowball in the beginning, I ended up with a rectangular, red storm cloud instead, which is also pretty cool. It may also look simple, but it is more complex looking than the trials I had with higher dimension Julia Sets. Anyways, I hope you enjoy my print! Thank you for tuning in! https://www.marksmath.org/visualization/julia_sets/ https://archive.bridgesmathart.org/2010/bridges2010-247.pdf Devaney pages 268-272, 295-300, 311-319. (Complex Analytic Dynamics)