Fractal Dimensions of Geometric Objects
In the last section, we learned how scaling and magnification relate to dimension, and we saw that the dimension, D, can be seen as the log of the number of pieces
divided by the log of the magnification factor. Expressed as an equation, we have D = log(N) / log(r).
Now let's apply this idea to some geometric fractals. We'll examine the Koch Curve fractal below:
As we learned in Chapter 2, geometric fractals can be made by starting with a simple generator pattern and replacing every section of the pattern with a smaller copy
of the generator. Let's look at the way the length of the curve changes as we iterate the fractal.
The generator (order 1) is made of 4 sections, and each section is 1/3 of the length of the initiator (order 0), which has a unit length of 1.
The second order of the Koch Curve has had each of the 4 sections of the generator replaced with the same shape, so it has 16 small segments, and each segment is 1/9
of the unit length, That means the total length of the second order curve is 16/9.
The third order curve follows the same pattern, and it has 64 tiny segments, each of which is 1/27 of the unit length, making a total length of 64/27.
As the progression continues, the curve gets longer and longer, and eventually becomes infinitely long! Now, it is not very useful to know that a curve is infinitely long,
and this is where the concept of Fractal Dimension becomes very useful.
Remembering that D = log(N) / log(r), we can calculate the dimension D by seeing how the number of units, N, changes with the magnification factor, r.
In this case, we can see that the number of pieces in the generator, N, is 4, and the magnification factor is 3, because each section of the generator is 1/3 of the unit length.This same relationship holds
between each of the orders of the curve. Order 4 has four times as many pieces as order 3, and each piece is 1/3 the scale.
So according to the formula D = log(N) / log(r), we can say that D = log(4) / log(3) = 1.26
Use a calculator (or Google) to find the value for log(4): [ ]
Find the value for log(3): [ ]
Verify that log(4) / log(3) = 1.26
But what does this mean?!
We're used to dimensions that are whole numbers, 1,2 or 3. What could a fractional dimension mean?
Fractional dimensions are very useful for describing fractal shapes. In fact, all fractals have dimensions that are fractions, not whole numbers.
We can make some sense out of the dimension, by comparing it to the simple, whole number dimensions. If a line is 1-Dimensional, and a plane is 2-Dimensional, then
a fractional dimension of 1.26 falls somewhere in between a line and a plane. And this describes the Koch Curve - it's wigglier than a straight line, but it doesn't fill up a whole 2-Dimensional plane either.
As we'll see soon, the more of a plane that a fractal covers the closer its dimensions is to 2.
Next, we'll determine the dimension of the Sierpinski Triangle.
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