Chapter 8: Images of Chaos

This next chapter deals with a complex concept which I have not encountered thus far in my education. I find it really interesting and will do my best to explain the idea. First, the complex plan is examined. In a complex plane, numbers from negaive infinite to positive infinite lie on a line stretching from east to west. These are real numbers. There is also a north-south latitude made up of imaginary numbers. These complex numbers are written in the following manner: 2+3i, I signifying imaginary number. Historically, imaginary numbers were invented to answer complex questions, such as “what is the square root of a nrgative number?” Then, the square root of -1 is i, and the square root of -4 would be 2i. This is an example of a complex plane, which is a geometric representation on the complex numbers in the real axis and in the imaginary axis.

In the complex plane, a complex object, the Mandelbrot set is found. TheMandelbrot, named after Benoit Mandelbrot set is a fractal. A fractal is a rough geometric shape which displays self-similarities at various scales, meaning it can be split in

to parts which are all reduced-size copies of the whole object. To graph the Mandelbrot set in a complex plane, first we need to find numerous numbers in the set. To do this, the equation, Z = Z² +  C, is used. Where C is the complex number corresponding to the point being tested, and Z is a number starting at zero. Only the magnitude of Z is used, and not the direction. To calculate themagnitude of a complex number, as add the square of the number’s distance from the x-axis to the square of the number’s distance from the y-axis, and take the square root of the result. As many complex numbers are tested, we can begin to graph the Mandelbrot set.

When thousands of points are plotted an image such as this one with form:

 

After taking time to understand this new idea, I found it really fascinating. There were some aspects of its make up that I didn’t fully understand, but for the most part the concept is not too complex.