Friday Fractal XLVIII: The Basics of Fractals

Today, you can create your own fractal. (Don't worry, I'll still include one of my artistic fractals at the end of this post.) You don't need to download any programs, or learn any new techniques. In fact, the only thing that you need is probably already sitting on your desk: a single piece of paper. (It can be any size, fresh or scribbled on; it doesn't really matter.)

Before we begin, take a look at the surface of the paper. How many dimensions does it have? (For this exercise, we're just looking at the surface, disregarding the edges.) Like the image here on your computer screen, it is 2-dimensional, that is, it can be measured in two directions (left/right or up/down.) Solid objects, on the other hand, are 3-dimensional, and can be measured in three directions (the same as before, plus volume, or front/back.) So far, so good, right?

Now, lets make that fractal. Take your paper, and wad it up into a ball. (Here's hoping you are using scrap paper, rather than a wedding picture or birth certificate.) Now, look closely. How many dimensions does the surface have now? It isn't a flat, 2-dimensional surface anymore; that much is certain. But, does it qualify exactly as a 3-dimensional surface? The surface of a solid sphere can be measured in only 2 dimensions, like using coordinates on a globe. The surface of our paper ball, on the other hand, folds down inside, embedded within three dimensions. Measuring it, we could easily determine the diameter, and thus the radius, of the ball. If the ball was solid, the volume would follow naturally. (V=4/3¶r) But it isn't quite solid, is it?

So, the surface has more than just two dimensions, but isn't quite three... so, what is it? The surface of the paper ball has a dimension somewhere in between, about 2.5. In other words, it has a fractal dimension. (If anyone out there is really into graphing things, go ahead and calculate this for yourself. Measure the diameter of the ball, and call the mass 1. Then uncrumple, and tear it in half. Make a new ball out of the half, calling its mass 1/2. Repeat, until you have no paper left. If you plot the measurements on a log-log graph, and the points appear to follow a line, the slope of that line is the fractal dimension of your paper ball.)

Now, we've just examined one of the first definitions of a fractal. Fractals exhibit fractal dimensions. Another basic property of a fractal can be seen if you uncrumple your paper ball, and examine the patterns of cracks and creases. Fractals exhibit self-similarity, that is, patterns repeat themselves at smaller and smaller scales. Your paper is likely filled with thick folds, joined by increasingly smaller creases. In between, the paper almost takes on a fuzzy appearance, as the creases become too small for the eye to see.

Many things in nature exhibit these types of patterns, rather than the rigid lines, shapes and solids we learn about in grade school. Sometimes, I try to recreate those patterns with a fractal generating program on my computer. (The 2006 collection can be seen here, while more recent examples can be found in the archives.) Other times, inspired by something as menial as a paper ball, I indulge myself in more artistic forms.

Like this:

The Fractal Depth of Paper

A spherical fractal with questionable dimensions

As always, check back for a new fractal, every Friday!

Fractal made by the author using ChaosPro