An Introduction to Fractal Nature

An Introduction to Fractal Nature

I don't claim to be anything approaching an expert on fractals, but I'll try and explain here what little I know. I shall confine this article to matters of self similarity and levels of detail. This is just an introduction to fractals, and not a tutourial on programming them. Anyway, lets get on with it.

A New Geometry

Elucidian Geometry is the stuff we all learn in school. It is the geometry of lines, planes, circles etc. It's simple and it works, and for a long time, mathematicians thought it was a resonable representation of nature. However, people soon discovered that they could draw (or at least begin to draw) certain curves and surfaces that could not be described by the origonal geometry.

What manner or curves are these?

How hard can it be to draw a curve? Let me attempt to describe one. This is the Koch curve:

Draw a triangle.

If we say that each line is of length 1, then the total length of the curve is 3.

Now take each edge in turn and add another triangle, a third of the size. So now there are 12 edges and 12 points. The length of the curve is now 4.

Repeat the process again, and again, forever.

length = 5.3333
length = 7.1111
length = 9.4815
length = 12.6420

As you continue adding edges, the length of the curve increases. If you add edges forever, then length of the curve reaches infinity, but the whole curve nevertheless covers a finite area. The curve is infinitely detailed. No matter how closely you zoom into the image, it always shows up more detail.

Self Similarity

So what do these mathematical curiosities have to do with the real world. Well, everything as it turns out. Such objects turn up all the time in the natural world. Animals, plants, rocks, crystals and liquids all exhibit fractal properties and self similarity.

Lets take a look at a common plant, the fern. The fern is typical of many plants in that it exhibits self similarity. A fern consists of a leaf, which is made up from many similar, but smaller leaves, each of which, in turn, is made from even smaller leaves. The closer you look, the more detail you see.

This is a standard fern, which you may well find while being dragged on long walks in the country by your parents long before you're able to fully appreciate the beauty of nature. You will see the overall theme of repeating leaves. Each smaller leaf looks similar to the larger leaf.
Looking a little closer, you can see that those small leaves are made up from even smaller leaves.
Closer still, and finer detail is visible.
No matter how close you look, more detail is always aparent.

Of course, in reality, a fern does have a smallest leaf, though I'm sure every fern aspires to be like that one. What is interesting it that the program to generate this image is only a few lines long. The same tends to be true for all fractals. A very simple algorithm can explain an infinitely complex object.