The Mandelbrot Set is proof that mathematics has the power to shine with beauty. When you look at a picture of a plotted Mandelbrot Set, you will see a black bulb that has many smaller circles and branches coming out of it. If you were to zoom in on any portion of it, all you would see is more and more circles and baby Mandelbrot sets. However, depending on where you zoom there can be other interesting patterns to see as well.
I created the images uploaded here using a Java Application that I wrote about two months ago. The mathematics behind its generation is complicated, but I will do my best to explain it briefly. What you are looking at is a set of complex numbers. The horizontal red line is the real number line (Numbers like 1, 2, -3, 2.7, or even pi). Anything above or below that red line is a complex number (a number that consists of a term with the square root of negative one). Complex numbers are written out with a real number (symbolized with A) being added to or subtracted by a complex coefficient (symbolized with BI where I equals the square root of negative one). This simplifies to A + BI.
To determine if a complex number is in the Mandelbrot Set, the program will put that number through a predetermined function over and over again. This can cause one of two things to happen. The first possibility is that as that complex number goes though the function over and over again, both A and B will get closer and closer to zero. If this happens, then the number is considered to be within the Mandelbrot Set, and the program will color that number black. The second possibility is that A and B will explode away from zero, and the number will bet further and further away each time. If that happens, then the number is not considered to be within the Mandelbrot set, and will be colored a shade of red depending on how slowly A and B explode.
Because the program has to put a number through a function a large number of times to create these images, it can take a few minutes to even an hour to generate them depending on the quality of the image and the computer. All of the images here were rendered in 8K quality (7680 pixels x 4320 pixels), but were done on a monster computer; they each took about a minute to generate.