Tutorial Chapter 5
Similarity DimensionTHE SIMPLEST METHOD
One way to calculate fractal dimension is by taking advantage of self-similarity. For example, suppose you have a 1-dimensional line segment. If you look at it with the magnification of 2, you will see 2 identical line segments. Letís use a variable D for dimension, e for magnification, and N for the number of identical shapes.
Now take a 2-dimensional square and a triangle. With the magnification of 2, you get 4 identical shapes in both of them.
Finally, take a 3-dimensional cube. Again, magnify it 2 times. Now, you will get 8 identical cubes.
With these three examples, you should see a clear pattern. If you take the magnification and raise it to the power of dimension... you will always get the number of shapes! Writing it as a formula, you get:
eD = N
Since we are trying to calculate the dimension, we should solve this equation for D. If you are familiar with logs, you should easily find that
D = log N / log e
If you are not familiar with logs, donít worry. You can find a "log" button on any scientific calculator.
HOW WE CAN USE IT
Using this formula, we can now calculate fractal dimension for some fractals. For the Peano Curve, you see that in the initial shape there are 9 identical line segments (N = 9). Each of them is 1/3 the size of the initial line segment, so the magnification is 3 (e = 3).
Using the formula, we find that D = log 9 / log 3 = 2. Since the final shape is a square, this is exactly what we expected! Now, take a look at another fractal, called the Koch Snowflake. In it, you can see four identical snowflakes (N = 4). Each of them is 1/3 of the entire snowflake, so e = 3.
Calculating the fractal dimension, we get: D = log 4 / log 3 = 1.26. The dimension is a fraction ó something you can never see in standard Euclidean geometry.