My infatuation in fractals began freshmen year at Greeley after taking a Seminar with one of the seniors. I’m not sure exactly when simple interest turned to a kind of obsession, but during that lesson something seemed to click. It seemed as if this was the universe’s answer to everything; the mystery was solved, however complex the answer was to understand. I’m still not sure if I was misunderstanding the lesson, or if I had somehow seen it for what it really was; a pattern to describe the way the universe works.
Nevertheless, that day followed me, and I tried to understand more about fractals through the resources I already had at my disposal-- through courses I was taking. Sophomore year, through my European History and Architecture courses, I learned about many ancient architectural feats-- Stonehenge, the Pyramids of Giza, the Parthenon, many Gothic Cathedrals, and the Taj Mahal-- and that they all somehow involved the use of the golden ratio. I will come back to how this relates to fractals later in the article, but for now know that each of these buildings use different aspects of their design to form the golden ratio. I was intrigued by the fact that fractals, what seemed to be something only formed by the forces of nature, were being constructed by human hands. Although I wanted badly to find out more, I waited until that summer, when I discovered a YouTube account by the name of Vihart. Vihart’s videos are not tutorials on how to do math, however Vihart’s ramblings about the nature and the concepts of the mathematical world have a lot of educational value, especially on topics that are more complicated to understand then to compute. Her videos on fractal math and their comparability to nature, helped to show me that there was a lot more to fractals than I had originally thought; I was hooked.
However difficult they are to understand conceptually, they are easily identifiable. Described as, “infinitely complex patterns that are self-similar across different scales,” (Wolfe) you are able to see when something looks like a fractal. As you zoom in on a picture of a fractal, you find that each image is the same (or similar) as the one preceding it. For example:
The matching colors represent the magnification of that part, called the iteration, of this particular fractal. One thing that makes fractals special is that they have their own dimensions, fractal dimensions. Dimensions are the number of planes an object lies on, one dimension represented by a line (an x-axis for example,) two dimensions represented by a square (an x and y-axis,) and three dimensions by a cube (or x, y, and z-axis.) Fractals fall somewhere in the middle of these standard dimensions because they are too detailed when on any plain to be called one, two, or three-dimensional (Luis).
Sierpinski triangle; each picture is an iteration of the one before it, removing the middle 1/4th of the of the whole (filled) triangle. (Luis)