Mandelbrot Set Variant

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Mandelbrot Set Variant George Mason University 401: Mathematics Through 3D Printing Mandelbrot and Julia sets are objects of study in complex analysis in which investigates the boundedness of complex numbers under certain iterations of complex-valued functions. Specifically, the Mandelbrot and Julia sets are of great interest even in pop culture due to certain properties such objects have. For example, the Mandelbrot set is self-similar at certain points of the set, giving the Mandelbrot set a fractal-like appearance which makes sense considering it is a fractal (although it has dimension 2. The Mandelbrot set refers to the set of all complex numbers c such that starting at z_0=0 the iterated function z_n+1=z_n^2+c remains bounded after infinitely many iterations. In this case, I have instead created a Mandelbrot set but instead uses the function z_n+1=z_n^3+c and using the same initial condition of z_0=0. The print shown above represents at each layer the complex numbers c that remain bounded after a certain number of iterations. For example the top of the object represents all the complex numbers that remain bounded after 30 iterations while the bottom layer represents all the complex numbers that are bounded after only 1 iteration. The object shown is the result of such iterations and tracking what complex numbers remain bounded after each consecutive iteration. Something else of note: the set shown above does not represent the entire Mandelbrot set associated with the function z_n+1=z_n^3+c. The actual set would be impossible to create due to limitations and so there may be some numbers in the set shown above that are not actually in the set but that we cannot distinguish from complex numbers that belong in the set. As for printing, the only difficulty I came across is that the size of the object should be scaled accordingly since if the object is smaller than say even 30mmX30mmX30mm the quality of the layers will decrease significantly and will not print as precisely as one would like.