What is FractalIS?

What are fractals?

You'll notice that the Z is on both sides of the equation. How could this be?!? That's against the rules of algebra! Well, this isn't algebra, this is Calculus. So the Z represents a dynamic complex number and the C represents a static complex number. Now I've told you that this is Calculus, but I didn't tell you why. In calculus, if a number is on both sides of an equation this means (in most cases) that this is an iterative function. An iterative function is an equation that feeds itself given a starting value. Here's an easy analogy:

Alright, this is easy enough... This equation simply says that the variable X is equal to itself plus one. So if X starts out equaling zero, after the first iteration X is equal to 1 and after the second iteration X is equal to 2..... etc. (get the pattern?). Well, it's pretty easy to say that if you were to iterate this equation until infinity, X would equal all integers eventually. Now let's say that you didn't want X to go off into infinity but you wanted it to stay within a certain cap (or amount that you want it to never go over). We can do this by adding some more to the equation. Let's say that we wanted the X variable to start at 30 and we wanted the absolute value of the number to approach zero randomly from there:

This equation would mean that X would start at 30 at first, and then it would equal the cosine if itself plus the sine of itself times itself. Here's a Graph:

Iterations	X
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20	-25.0134052261583 -27.81342742228 37.2813925929243 18.9516118285601 20.7837713088013 12.038269043087 4.33222127861861 -5.63049537897767 -7.89270812308245 8.19237083377528 5.0081810495381 -3.33081072206091 2.64486566827669 -1.06481337047388 .415310250728614 .547572035111863 .752586764121447 1.06374554390853 1.44646168711116 1.61467785947979

This is an example of pseudo-randomness in math. Now, The Mandelbrot equation is slightly more complicated (We won't ge into too many details). There are two parts to each variable Z and C. These consist of one real number and one imaginary number (the imaginary number system is a very complex one and we will not handle this here). Now as this equation iterates, it gets closer and closer to infinity (kind of like X = X + 1 except more random). Obviously, we can't let this thing trail off into infinity because we'd never see a pixel plotted. So we alot a value to the cap. Let's say the cap is 40, when X becomes equal or greater than fourty the equation stops iterating and moves on to the next plot. When I say the next plot, I mean the next pixel (if your screen size is 640 x 480 pixels, FractalIS has to iterate the equation for each pixel a maximum of 40 times if the cap is 40). So essentially FractalIS has to do about 12.288 million iterations of the equation just to give you a filled screen at a Max Iteration (cap) of 40. So, the black area that you see on the screen when you first start is the actual Mandelbrot Set. The rest of the colorful fancy stuff is just the steps that the equation goes through to get there. Theoretically, if we could have a cap of infinity, there would be no black area on the screen. This is because the more iterations you have, the more, "Fractal Resolution" you get and the closer you get to an actual view of the mandelbrot set. It's rather funny, Calculus, it seems that using calculus you never get anywhere. Everything is infinite and you can never reach a destination that you want because it's not the actual destination that you're looking for. A good analogy is: Let's say that you are told that your mother will be killed unless you walk from one end of the room to the other. The only constituent is that every step you take, you shrink to 90 percent of your current size (your stepping is the iteration and your Size = Size * .9). Well, to tell you bluntly, you're never ever going to get to the other side of the room because relatively to you, you'll still be using the same stride, but relatively to the room, your stride gets smaller and smaller until it seems that you aren't even moving at all! Let's hope that you never end up in this situation. Well, this is kind of like what it is to explore a fractal. You'll never get to the end of the rainbow because every time you zoom in, your reference compared to the original size of the fractal gets smaller. This concept raises an interesting topic.... Infinity of size. The concept in a nutshell is that everything in the universe is infinite in size. Inside of the universe are Galactic clusters, inside of those are galaxies, in those, are solar systems, inside of solar systems lie planets, inside of planets (earth) lie objects and lifeforms, inside of lifeforms are cells, cells are made up of many things including molecules, inside of molecules are atoms (which are no longer deserving of their name by the way), atoms consist of a nucleus and electrons, inside of electrons are quarks, inside of quarks are..., why should we stop here? Is it because that's all that we can see? Many think that we shouldn't stop here. Let's say that past quarks are some other building block, and inside of those are building blocks of the building blocks of quarks, etc... Conversely, we can assume that the universe is a building block of something else and so on.
According to the laws of chaos, somewhere down past the levels of quarks, we should run into something that is roughly like our own universe. If you don't believe me, just download FractalIS and see for yourself. One of the main concepts of Chaos Theory is self similarity. This means that once you get so far, you begin to see something that looks like the original, but with slight or drastic differences (depending on the size of your frame of reference). When you begin to explore the Mandelbrot fractal, you notice that there are an infinite amount of "little" Mandelbrots (often called minibrots) that fill your screen with a dazzling array of colors and diversity. These minibrots are somewhat like the image that you see when you first start, but they can be decieving, because every minibrot is completely an original with it's own minibrots in different orders than every other minibrot ever (see if you can say that 10 times fast). This is the wonderment that is self-similarity.

And so concludes my FractalIS introduction, the rest of what you learn is up to you.

Have fun, and remember, The universe is math, Art is our interpretation of it.

~TaxDAY