The Mandelbrot set is a set of points in a complex plane, c, of the orbit, which is how the function operates under the iteration of zn+1 = zn2 + c around 0.
Mandelbrot Set
The Mandelbrot set is a set of points in a complex plane, ‘c’, of the orbit, which is how the function operates under the iteration of zn+1 = zn2 + c around 0. A value of ‘c’ is included the Mandelbrot set if the orbit of 0 undergoing the iteration of zn+1 = zn2 + c, and the value s does not tend to infinity. In other words if the orbit of 0 tends to infinity, then that the ‘c’ value is not in the set.
To see this properly let ‘c’ be any complex number and then let zn be 0 in the iteration of zn+1 = zn2 + c, you will notice that you will get c back from the resulting answer; 02 + c. This can repeated by letting c be x for the next iteration in the original equation, and this will yield c2 + c. You can continue repeating putting the previous answer in for x into the equation and the result will be (c2 + c) 2 + c. Doing this continuously will create a list of complex numbers and if these complex numbers are increasing in size and thus further away from the origin then ‘c’ is not in the Mandelbrot Set. If however the opposite occurred and it not move away from the Mandelbrot set and did not increase in size then it is considered to be inside of the Mandelbrot Set. Some examples of numbers in the Mandelbrot Set are: 0, -1, i, and -2. Below is an example of four iterations of the c value -1.
