Fractal Theory: The Battle for Z's Destiny
Z starts out being initialized to zero, that is, both the real and imaginary parts are set to zero. C is initialized to the complex number representing the point to calculate, its real portion is its horizontal distance from the center of the plane, its imaginary portion is its vertical distance from the center of the plane. Continuing to go through the loop (described in the previous panel), take the square Z and add C to it. What's the point to all of this? Whether Z will start getting large, or will stay a very small number, trapped around the center of the complex plane.
After the first iteration, Z is equal to C, because zero squared is zero. Then, if Z is larger than one, when it is squared, it will leap outwards, trying to break free. However, if C is pointing in the opposite direction, then when C is added in, it will pull Z back. If Z is smaller than one, squaring Z makes it even smaller, but again, C is a wild card -- which way will it push Z -- in or out? There is no shortcut way of finding out. The calculations have to be performed. But as we zoom in to the convoluted boundary of the Mandelbrot set, we can see that Z and C have fought a mighty battle to see whether Z escapes or not. Constantly changing sides, teetering near the brink of two, only to fall back towards zero. It is a battle where a change of a millionth of a unit can be the difference between staying forever trapped or shooting off to infinity.
The Julia set is very similar to the Mandelbrot set in how it is calculated. The actual calculation loop is identical, but the initialization is slightly different. For the Julia set, Z is initialized to the current point, and C is initialized to a seed value, another complex number which is typically taken from the Mandelbrot set. For each different value of C, you get an entirely different Julia set. Because an infinite number of values for C exists, there are an infinite number of Julia sets, each of which can be zoomed to any level of magnification. That's why, a thousand years from now, you'll still be able to look around the Mandelbrot and Julia sets and know that you're almost certainly exploring an area that no human being has ever seen before.