While I was doing research and notes for my independent assignment, I realized just how deep and complex the field of fractals is. I took the decision of selecting fractals as the main topic quite lightly, mainly because they are just so fascinating and mysterious! However now I realize that truly understanding them is not for the slightest faint of heart. I found some amazing website explanations of fractals, and lets say, some not so great. I don’t blame them though, it is in fact really hard to explain “imaginary complex numbers”, and I don’t pretend to understand them myself.
You see, the Mandelbrot Set, which is the most popular and in my opinion also the most beautiful, is based on an iterative equation . Being that the equation is iterative just like all fractals are, this means that a given number must be put into the equation many times, in order to get the final result. Sounds simple, right? Well in reality, many calculations occur before and throughout this process, which I will try to explain in more detail here.
Basically, a fractal integrates complex numbers into its equation, a number that cannot possibly exist in reality. So how can it exist if it can’t exist? Well, this is where it gets tricky, and the number i will haunt you just as it did me. The number i was created because no real number can be squared and give a negative number as a result. This number (umm, letter) is equal to the square root of -1, which is not possible, but then again, what number squared would give -1? We don’t know, so that’s why we represent it with the letter i.
The imaginary unit i is used in complex numbers in the complex form a+bi, in which the a is the real part graphed on the x-axis, and b multiplied by i is the imaginary part, which shows up on the y-axis. In fractals, we are not concerned about the actual complex number, but its magnitude, or distance from zero. with real numbers, this is simply the absolute value of that number. However, the magnitude of a complex number is its distance from the origin of the Cartesian plane, which is found by taking the square of the distance from both the x and y axes, then taking the square root of their sum.
Example: If you are given the number 5i, and you want to know its square, simply square the real part, and then change it to negative, giving -25. Similarly in the expression 3i, the square would be 9 negative, because 3 times i would be the square times -1, which is -9.
Now on to the fun stuff, actually graphing the fractal BY HAND! To start off, draw a Cartesian plane that extends to 2 in all directions, since that is the size of the fractal. The horizontal axis can be labelled RE, for real, and the vertical IM, for imaginary, and both should be scaled to at least 0.5 (giving a 9×9 grid, which would be sufficient for a rough sketch, a smaller scale would be even better, but more time-consuming). Each square would be given a coordinate, such as (-2, 2), (-1.5, 1), (1, 0.5) etc, which will be substituted into the equation to determine if it surpasses 2.
The process of selecting a coordinate, determining its magnitude, substituting into the equation using the complex number Z and the constant C, and iterating this equation to determine whether the point belongs to the Mandelbrot set, will be explained in more detail in my next blog post.