Fractals

In the course of your mathematical studies, you may have come across strangely beautiful images of fractals, as in the picture above. Many mathematicians have studied topics related to fractals, and the basic ideas behind them have occupied mathematicians for millennia. However, the proper study of fractal geometry as a single topic is a fairly recent development. Credit for the development of fractal geometry belongs most to Benoit Mandelbrot, who invented the term fractal in 1975. Aside from being weirdly attractive mathematical objects, fractals have many applications in computer science, and a great many fractal forms can be found in nature.

Fractals have three main characteristics. First, they are self-similar shapes. This means that small parts of the shape are similar to the larger shape. Second, fractals are formed by the infinite repetition of a mathematical process: each individual repetition of the process is called an iteration. (The word "iteration" comes from the Latin word iter, meaning "a going" or "journey". We use words similar to "iteration" when we speak of an "itinerary" for a vacation or "reiterate" something we have said before.) The iterations of a fractal are infinite in number, because the output of each iteration is made input for the next. Third, and very importantly, fractals have fractional dimension (this is why they are called "fractals"). Strange as it may seem, a fractal shape can have a dimension of, say, 2.4, rather than just 2, or just 3. (You may want to investigate this further on your own.)

Once you recognize the characteristics of fractals, you can begin both to find fractal forms in nature and to find natural forms in fractals. A fern, for example, is self-similar: any branch or leaf on the plant replicates the design of the plant as a whole, and it is possible to write a mathematical process which closely resembles a fern. More generally, the branching patterns of trees have a kind of fractal form. If you look at a tree that splits into three branches from its trunk, you will probably see that each branch then splits into three smaller branches, and so on. Rivers and their tributaries follow such patterns, as do the blood vessels in your body. Mountains, too, can exhibit fractal patterns, since the smaller peaks and valleys they contain replicate the shape of the mountain itself.

Fractals have the interesting property, also, of seeming simultaneously finite and infinite. The best explanation of this comes from Mandelbrot himself, who uses the example of the British coastline. Mandelbrot points out that if you take measurements of the coastline of England from the air, you might say that it has a length of, say, several thousand miles. However, if you were to measure the coast in meters, you might find your result to be considerably longer, because of the coast's many twists and turns. You could continue this way forever: the smaller the scale of measure (going down to the infinitely small), the larger will be the measurement (going up to the infinitely large). Yet it is always the same coastline!

Computers are particularly suited to the study of fractals. The huge number of iterations necessary for a proper investigation of fractal forms is beyond the ability (and patience) of a human--but if computers can be said to have a talent, it might best be described as the ability to do the same thing over and over and over again (to iterate and iterate... and then reiterate until the thing is done). For this reason, computers are often used to simulate natural fractal patterns, whether they are in landscapes, blood vessels, cloud formations, weather patterns, or other natural forms. Two fractals you might want to investigate further are the Sierpinski Triangle (or Sierpinski Gasket) and the Mandelbrot Set.

Things to Think About...
 1. What is the Koch snowflake?
 2. What examples of fractals can you find at home?
 3. How can star clusters be fractals?