How does this relate to the Mandelbrot set? The Mandelbrot set is similar in many ways but far different in just as many. To begin, the Mandelbrot set is a fractal and is a repeating pattern with infinitely small detail. Basically you could zoom in on a Mandelbrot set forever and never reach the end. The Mandelbrot set also is created by a simple principle. The set is made by taking all of the numbers that don't increase or decrease without bound and fit the equation Z = Z2 + C. C can be any number in the complex plane.
One more quick detor before we finally dive into the Mandelbrot set. The Complex plane is where all those unreal roots of quadratic functions go to die…or at least that's what I like to believe. It is a plane where, instead of x and y, the axis are made of a+b(i) where "i" is the square root of one. "a" and "b(i)" represent the two axis respectively and create the plane we call the complex plane. If your head just exploded, don't worry mine did as well.
So what exactly is the Mandelbrot set? Well, we know it is a fractal created by the equation Z = Z2 + C on the complex plane, but what exactly does that mean? Z is a number that changes as we repeat the equation infinitely. Z starts off as zero and C is what we pick to put into the set from the complex plane. C can be any number in the complex plane that satisfies the equation infinitely as well as one other important thing. The value of Z cannot increase or decrease without bound. It must stay "near" C in order to make it into the Mandelbrot set. The Mandelbrot set is the collection of all numbers that satisfy this equation in graph form. The Mandelbrot set is not the equation Z = Z2 + C, but rather the graph made by taking all of the points that satisfy the previously stated conditions. Z = Z2 + C is actually a set known as the Julius set where the Mandelbrot set is the representation of all Julius sets rolled into one. To put is simply, any number that would be in the Mandelbrot set has its own Julius set. The Mandelbrot set is all of them.
So just what is this thing we've been talking about? Here it is, for your amusement, the Mandelbrot Set:
The Mandelbrot set has one other very important difference that makes it unlike other fractals. While you can zoom in on parts of the Mandelbrot set to see a repeating image, you can also find images that are not repeated or repeated fewer times than others. The Mandelbrot set has varied images within it that share little to no resemblance to the original shape.
This is a picture from a progression deep into a valley on the Mandelbrot set and it is easy to see why it is so complex. The image here also shares an almost unsettling resemblance to a splotch of organic fluid on a slide of a microscope. Why is this? Why does the Mandelbrot Set have patterns that the shape as a whole does not share? How has is this infinitely complex function created by a such an unassuming equation? Perhaps the greatest complexity the Mandelbrot Set has to offer us is own fascination.