Compare And Contrast Mandelbrot And Filled Julia Set

The Mandelbrot and Filled Julia Sets The Mandelbrot and Filled Julia Sets are sets of numbers that are composed of complex numbers that do not diverge for varying conditions of the function fc(z)=z²+c. For the Mandelbrot set, the function is iterated from z=0. The filled Julia set holds c constant for any complex number to look at the behavior of the iterated functions for each value of z in the complex plane. Both sets are derived from the iterations of simple functions in the complex plane, but cause unpredictable, infinite fractal patterns when graphed. Furthermore, the sets are integral parts of chaos theory and hold a number of other mathematical truths within them, such as the value of pi.
The groundwork for both functions was set by …show more content…
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 …show more content…
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