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Do you eat broccoli? If so, did you know you were eating a fractal? Don't worry, it's not something dangerous. A fractal is a kind of pattern, one that is created when an action is repeated on smaller and smaller scales within an object. Mathematicians refer to a fractal as a self-similar pattern, created by a repeating operation (or iteration), having fractional dimensions. That's a fairly complicated explanation, so let's take a look at this definition piece by piece.
When you look at a fractal, you see a large shape that contains
smaller ones that resemble it. This is called self-similarity. Imagine
a rocky mountain. From a distance, the mountain has a rocky or uneven
shape. Up close, you see that the smaller rocks that make up the
mountain also have uneven shapes, just like the overall mountain.
There are many other examples of fractals in nature. The next time
you take a nature walk, look at a tree very closely. If you look
at a branch of a tree, you will notice that it looks similar to
the whole tree. This same self-similarity is present in the
system of blood vessels in the human body and in a stalk of broccoli.
An iteration is an operation that can be repeated. Think
again of a tree or of a stalk of broccoli. First, the trunk or stalk
grows, and as it grows, imagine that 3 limbs grow from it. Then,
3 branches grow on each of these limbs, producing 9 total branches.
After that, 3 twigs grow on each of the 9 branches, producing 27
twigs. The number of new branches can be described mathematically.
If the pattern is 3, 9, 27, and so on, you can describe the pattern
mathematically with the expression 3n. In this expression,
n stands for the number of iterations.
The operation in our tree branch example is the actual growing
of 3 branches. It is the process of creating an iteration. For example,
in a geometric fractal such as "Cantor dust," you "operate" on a
line by taking out the middle portion of the line with each step.
(See your book, p. 611, for pictures of these fractals.)
What about dimensions? It is not enough to produce a self-similar figure. Each new figure must also be of a certain dimension or size. In Cantor dust, each line segment produced is one-third the length of the original line. The new segment is a "fraction" of this original figure. This is why mathematicians describe fractals as having "fractional dimensions."
Why study fractals? Well, mathematicians are always looking for
new ways to describe what they see using math. For centuries, humans
have looked to the line as the basic structure of the world around
us. Fractal geometry gives us a new way to approach and understand
nature. It's like a new language that can be used to describe complex
actions in nature, such as water flow, the behavior of various animal
species, and the growth of broccoli.
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