Julia set for z^2+z+(-1+.2i)
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Julia Set For for z^2+z+(-1+.2i) By Krista Cimbalista 10/28/2020 Math 401 Mathematics Through 3D Printing GMU NOTE: you will need to scale the STL file in the z-direction by a LOT. Mandelbrot sets and Julia sets are typically generated by the equation f(n-1) = x(n)^2 + c. For the Mandelbrot set, your initial value is zero and you pick a complex number for the constant and iterate. For the Julia set, you choose both an initial value and a complex constant to iterate over. The sets that are generated are appealing and intricate with many different designs to visualize and appreciate [1]. Mathematica makes generating these sets very easy with commands like MandelbrotSetPlot and JuliaSetPlot. Something worth explaining is the coloring of the 2D graph. As you can see in my Mathematica image above, there is a chart to the right of the 2D JuliaSetPlot. What this is showing us is the number of iterations it took before we could be sure that each of these points was not in the Julia set. The higher the iteration count (the black/white areas), the more iterations were needed, if we even iterated enough, to know that it wasn't in the set. The lower the iteration count (the blue/purple areas), the sooner it was clear that these points were in the set. You can see that the black regions on the 2D plot correspond to flat regions on the 3D plot, this is because those areas needed more than the 100 iterations calculated to tell us if those points are a part or the set or not. Of course, any polynomial can be used to generate a Julia set but, in my case, I used a second degree polynomial consisting of three terms instead of two. Working with this was a little harder, because it was not as straightforward to find a constant that was inside this set. Eventually a constant I found and liked was -1+.2i. This generated a Julia set that was longer than it was wide with lots of “decoration” around the edges and within the shape. I like to think that the shapes here look like little galaxies. To code this project, I first used Mathematica to plot and find a constant that was in the Julia set corresponding to my polynomial. Next, I made it into a 3D plot and exported it as an STL. Then, I imported that STL into OpenSCAD where I removed some of the bottom layers to make it printable. Sources: [1] https://www.youtube.com/watch?v=mg4bp7G0D3s&ab_channel=JimiSol Useful link: https://www.marksmath.org/visualization/julia_sets/
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