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