## 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.

*Uncategorized*

Posted on 30 June 19930