An important part of understanding fractals is the notion of fractal dimension.
We all learned back in Grade School Math that lines are one dimensional,
squares and circles are two dimensional, and
cubes and spheres are three dimensional.
These perfect geometric objects are often what mathematicians and scientists use to model the attributes of the natural world.
However, as Benoit Mandelbrot famously said,
"Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth,
nor does lightning travel in a straight line."
Mandelbrot proposed that fractals are a much better model of the natural world than our more conventional, geometric notions
and he sought to develop a new, fractal geometry to describe nature.
To develop this new geometry, we need to examine our concept of dimensionality.
Let's look at what exactly we mean by our ordinary concept of dimension.
Our ordinary notion of dimension is the extension of an object in a given direction.
For example, a line is one-dimensional. It extends in a one-dimensional space.
A square is two-dimensional. It has two directions of extension.
And a cube is three-dimensional, with three directions of extension.
But where should we put something like the Koch Curve?
Well, it's made up of straight lines.
You can think of it as being sort of like one-dimensional.
You might imagine stretching it out.
But the problem is, the more levels we go down in constructing it, you know the longer it gets and
as you go down an infinite number of levels its length becomes infinite.
So all of these little bumps and valleys and so on make it somewhere between one- and two-dimensional.
And I'll show you why that is.
That's a very strange notion, that something could have fractional dimension.
But when you really think about what it is we mean precisely by dimension, then it will make sense.
Let's look at dimension in a more mathematically precise way.
Here's a way of characterizing dimension.
Look at what happens when you continually bisect--that is cut into two equal halves--the sides of lines, squares, cubes, and so on.
Take a line--this line right here. My next step--indicated by this arrow--is to cut it in half.
So I cut the line into two equal parts.
And then I do the same thing in an iterative process.
I take this sub-line, cut it into two equal parts, and
take this sub-line and cut it into two equal parts, and so on.
I can do the same thing for the square.
I can take this side and this side, and cut them into two equal parts, and I get four sub-squares.
I do the same thing and take each of their sides and bisect them and get
four more sub-squares of that original sub-square, and so on.
I can keep doing this.
Here's my cube, where I've taken each line here and bisected it, and bisected it, the same thing up here.
You can see how this is an iteration--a kind of building of a fractal, if you will.
But now we can start counting, and
we can look at, for our line, a one-dimensional object.
When we bisected the line at each level, we saw that
each level is made up of two, one-half-sized copies of the previous level.
For example, this level here is made up of two copies of the previous level.
These are both lines that are each half the size.
Similarly, over here, each one of these is one half the size of one of these, and there are two copies.
Well, if we look at the square, we see that each level is made up of four, one-fourth-size copies of the previous level.
Take this as the original level, and here we have one, two, three, four copies.
Four new squares, and each one is one-fourth the size of the original.
We do the same thing here. For each one of these original squares we have four copies, each one-fourth the size of the original.
And so on. And we can see, obviously, for Dimension-3, that each level is made up of eight, one-eighth-size copies of the previous level.
You can try this yourself for a four-dimensional cube.
Actually, that might be kind of hard.
You can see a pattern here, and I can let you know that each level of a four-dimensional cube, or hypercube,
is made up of sixteen, one-sixteenth-size copies of the previous level.
So now, your turn.
Suppose we have a twenty-dimensional cube. What is each level made up of?
Well, that's your question for the quiz.