Fractals: Exploring Mandelbrot Coordinates and qualitative characteristics of the corresponding Julia Set

(1) Hinsdale South High School, Darien, Illinois, (2) Westview Hills Middle School, Willowbrook, Illinois

The Mandelbrot Set and Julia Sets are fractals that are generated employing the cyclic equation z_{n+1} = z_{n}^{2} + C an infinite number of times, where z_{n} and C are complex numbers. For the Mandelbrot set, the value of z_{n} is static and the equation projects points that do not diverge to infinity for all possible values of C derived from coordinates on the complex plane. For Julia Sets, every value of C remains static while every value for z_{n} that does not diverge to infinity is projected. Because the standard, 2D Mandelbrot set assumes the value of z_{n} = 0, there is only one primary depiction of the Mandelbrot Set. For Julia Sets, the value of C is not assumed, and therefore, there are an infinite number of Julia Sets. In this study, the effect of changing the coordinate from which the value of C is derived in the complex plane for the equation z_{n+1} = z_{n}^{2} + C on the qualitative characteristics of the coordinate’s corresponding, standard 2D Julia Set in relation to the standard, 2D Mandelbrot Set is observed — specifically, Julia Sets derived from points outside the Mandelbrot Set, in the main cardioid of the Mandelbrot Set, in the primary bulbs of the Mandelbrot Set, along the edges of the Mandelbrot set, along the real and imaginary axes.

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