The groundwork for both functions was set by

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For many values of c in the Mandelbrot set, the iterations of the function produce outputs that oscillate between any number of values. For example, for c=-1, it cycles between 2 values. There are values that can cause any number in the cycle, with some values creating cycles of 3,4,5, etc. Additionally, it may tend towards a cycle, and not be a perfect cycle, but generally form a cyclic pattern even if the values aren’t constant. Obviously, for values of c not in the Mandelbrot set, there is no cycle, and the orbits are considered chaotic- as it constantly expands, there will be no cycling of

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Interestingly enough, the points at which the fractal pinches together to start a new “bulb” are connected to the Mandelbrot Set. For example, a special Filled Julia Set called the Douady Rabbit has a fractal that continually breaks into 3 pieces. The values of c that produce Douady Rabbits have a cyclic orbit of 3 in the Mandelbrot Set, meaning that they either cycle through 3 values or tend towards a cycle of 3. This is true for all Filled Julia Sets that fractures: The number of pieces it fractures into is equal to the cycle of its orbits in the Mandelbrot