A Short Exposition on Dimensions and Fractal Dimensions

Posted on 30 June 1993

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Thomas J. McFarlane

Dimensions

A dimension is a direction of freedom. We speak of space as
having three dimensions because there are three independent
directions in which we are free to move and because positions
in space can be specified by exactly three measurements. A
surface, such as sphere, has two dimensions because there are
only two independent directions of free movement upon the
surface. A line or curve has one dimension and a point has
zero dimensions.

Although we usually think in terms of physical dimensions,
the concept of dimension (as the number of measurements
needed to make a unique specification) can be generalized.
Mathematicians routinely work with abstract spaces of
arbitrarily large dimension, and even spaces of infinite
dimension.

Fractal Dimensions

The concept of dimension discussed thus far applies to
idealized objects such as points, lines, and surfaces. But
the objects we find in nature do not conform perfectly to
such idealizations. Take, for example, a human hair. As a
first approximation this hair has one dimension since, like a
curve, it has just one direction of extension. But if you
look at it in a microscope, you see that it is more
accurately described as being a three dimensional tube-like
structure. And if you analyze it further, describing its
atoms with quantum mechanics, you will find yourself in an
infinite dimensional Hilbert space!

Most objects are not even as nice as a smooth hair, though.
What about a tree or a rocky coastline? Such complex shapes
were largely ignored by science because there were no ideal
objects around that corresponded well to them. Then
Mandelbrot invented the idea of a fractal. (In fact,
fractals had been around for a long time before him. But he
was the first to give them a name, study them, and make them
popular.)

A fractal is an infinitely complex form that is characterized
by self-similarity, which means that a part of it is similar
to the whole. The most famous fractal is the Mandelbrot set,
and if you look closely at it, you can see little images of
the whole nested within it. But there are much simpler
fractals than the Mandelbrot set. For example, take a
straight line and make a bump in it, like this:

                  ___________________
_________________|                   |_________________

Then take each of the three lines and make a bump in them,
like this:

                         _____                
       _____       _____|     |_____       _____    
 _____|     |_____|                 |_____|     |_____

Then repeat again with each of these smaller lines:

                           _
         _           _   _| |_   _           _
   _   _| |_   _   _| |_|     |_| |_   _   _| |_   _
 _| |_|     |_| |_|                 |_| |_|     |_| |_

Now repeat ad infinitum!

Notice that this form is self-similar. Each little bit is a
copy of the whole. There are several amazing things about
this fractal. First, it has infinite length (since we’ve
added an infinite number of bumps). Second, it is so
“smeared out” that it is not really properly called a curve
anymore. Such curves have a fractal dimension between 1 and
2. A “smooth” fractal will have a dimension closer to 1,
while a very “jagged” fractal will have a dimension closer to
2. There is, in fact, a way to define the fractal dimension
that makes this notion mathematically precise.

There is a wonderful discussion of fractals (as well as many
other mathematical things) in Rudy Rucker’s book Mind Tools,
if anyone is interested in reading more.

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