Audri Swisher's Blog

“Hunting the Hidden Dimension” presents fractal geometry: the means of describing mathematically those patterns that appear jagged and random upon first glance. This is done by breaking a typical geometric shape into endlessly smaller and smaller pieces – “iteration” – with often surprising results. The key idea is self-similarity: that is, no matter what the scale is, a fractal will look very much the same.

Essentially, Benoit Mandelbrot flew in the face of classical mathematics with his discovery of fractal geometry. The long-held assumption was that math was limited to smooth lines and regularity. In the 1970s, Mandelbrot pointed to the seeming chaos of nature and claimed that fractals could describe its patterns, as classical geometry described man-made patterns.

In fact, fractals seem to be all around us – tree branch patterns, special effects of films, art, noise on telephone lines, textiles, antennas, cell phone parts, our eye movements, cancer screening, etc. While their full potential likely hasn’t been realized, it’s clear that fractals have already been beneficial to us. As for what lies ahead…who knows? (There’s probably a fractal pattern out there somewhere to predict it, though.)

Response:

So this is what the math teacher was talking about when he said that what we were learning was applicable.

While some of the people in this video are quick to point out that the technicalities aren’t the most important things about fractals, I was still intrigued by their explanation of being “in between dimensions.” According to mighty Wikipedia, a dimension is “informally defined as the minimum number of coordinates needed to specify each point within it.” All right, so locating a point in a two-dimensional square needs (x,y), locating a point in a three-dimensional cube needs (x,y,z). Therefore, a 2.6-dimensional fractal needs…2.6 points? Hmm. Oh well.

Trivialities aside, fractals are pretty cool. And definitely useful. But beautiful? Biologist Brian Enquist (University of Arizona) had to say: “What’s absolutely amazing is that you can translate what you see in the natural world in the language of mathematics. And I can’t think of anything more beautiful than that.”  I might not go that far, but it is fascinating to think that seemingly unrelated objects, such as a mathematical formula and our eye movements, could be linked somehow. I’m curious about what precisely he means, though. Is he amazed that nature fits these patterns? Or, conversely, that the patterns fit nature? Is he simply amazed at our ability to translate them? That simplicity results from seeming complexity?

Richard Taylor (University of Oregon) also said that mathematicians and artists are closer than a person might think: they just use different languages. At another point, Mandelbrot stated: “The eye had been banished out of science…been excommunicated.” He, on the other hand, was fascinated by the visual side of mathematics. From a pure visual standpoint, fractals are eye-catching and beautiful. But, as some of you have probably already figured out, I’m much more concerned with the practical side of things. If useful fractals are also beautiful, more power to them. If they ensure that my cell phone doesn’t look like a porcupine, even better.

Here’s a fun little site for playing with fractals: http://www.fractalposter.com/fractal_generator.php

“Why are numbers beautiful? It’s like asking why is Beethoven’s Ninth Symphony beautiful. If you don’t see why, someone can’t tell you. I know numbers are beautiful. If they aren’t beautiful, nothing is.” ~ Paul Erdős