WHAT IS A FRACTAL?
Gazing upon the image of a fractal is vastly more entertaining than the verbal unraveling of it. In case you have trouble falling asleep at night, below is the definition of a fractal and an explanation of how it creates chaos.
But first, my humble opinion…
God accidentally created the universe (chaos) in an attempt to create mankind with a freewill (i.e. a seemingly ordered self-rulership of divinely assigned molecules). If my theory is true, then when we gaze out at the awesomeness of the tiny part of the universe that we can actually see, we are observing a microscopic portion of the byproduct of our existence. I've heard that "in the beginning was the word..." I wonder if God's second word was "uh-oh".
I have traveled the outer reaches of inner space where I followed the path of the fractal until it all began to look the same, only smaller. Finally, growing weary of the predictable, I took the quantum leap of logic I seem to be known for, into the land of chaos, where water jumps back into the faucet, birds turn into dinosaurs, dirt sucks up flowers, and the sun takes back it’s shine.
That being the case, I wanted to tell you that I’ve recently amazed myself with the seemingly miraculous feat of willing every single molecule from my aura, inward, to move forward in space and time simultaneously and appear, as if by magic, in their previously recognizable form every time I pass my reflection in a mirror... with virtually no effort! Wow. I am easily amazed. However, I seem to be slipping on the molecular level as of late, because my re-assembled molecules are looking more and more like my mother every time. Mirrors should come with a warning.
Okay, the definition. Several have been created over the years as mathematicians struggled with the complex properties of fractals. Here they are:
Any shape that has the unusual property that when you measure its length, area, surface area or volume in discrete finite units, the measured value increases without finite limit as the size of the discrete unit decreases to zero.
Any object that is self-similar in a non-trivial manner
A geometric figure or natural object that combines the following characteristics: a) its parts have the same form or structure as the whole, except that they are at a different scale and may be slightly deformed; b) its form is extremely irregular or fragmented, and remains so, whatever the scale of examination; c) it contains "distinct elements" whose scales are very varied and cover a large range.
"Chaos theory" or the study of "chaotic" processes is related to the study of fractals because such processes will yield a fractal image if a certain attribute is plotted on a graph.
Such chaotic processes usually involve some sort of dynamic process (things changing over time) following seemingly simple rules, usually with no random element but nevertheless showing a seemingly "random" behavior.
A common technique in the study of chaotic processes is to look at what happens when a "parameter" is changed. Different values of the parameter will cause different types of behavior, such as convergence to a constant value, divergence to infinity, simple oscillation, more complex oscillation, or "random" non-repeating oscillation. When you make a color-coded plot of what happens as a function of the parameter, you often get a fractal image. This is how the Mandelbrot Set is plotted.
Sometimes a picture is worth 557 words.