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Fractals

Determining Fractal Dimension

We will see that the dimension of a Koch curve is not an integer! Let us start from a simple segment of line again. Divide it up into four equal segments, each of length 1/4. Let's call N the number of segments created and let us call R the length of each segment. Then, N is 4 and R is 1/4. Thus, the unit segment and its parts (the four smaller segments) are all similar, so this is a self-similar structure. Start now with a square of unit side and divide the square into nine squares of side 1/3. Here N=9 and R=1/3. The big square and its parts (the nine squares inside it) are similar; and so the structure is also self-similar. The same can be done with a cube, divide its side in three and obtain 27 cubes of side 1/3, or 8 cubes of side 1/2, etc. Hence for a cube, if N=27, then R=1/3, or if N=8, then R=1/2.



For all of these self-similar constructions one single relationship applies. N=(1/R)^D, where D is 1, 2 or 3 depending on the dimension of the object. A line is one dimensional, a square is two-dimensional and a cube is three-dimensional. So we can easily obtain a formula for the dimension of any object provided we can determine in how many parts it gets divided up into (N) when we reduce its linear size, or scale it down (1/R).

The formula is obtained by simply taking base ten logarithms and solving for D to get,

D=logN/log(1/R).

By applying this formula to the Koch curve, as we constructed it above, we see that it has a dimension that is no longer an integer!

Iteration number 1 2 3 4 5
Number of segments, N 4 16 64 256 1024
Segment length, R 1/3 1/9 1/27 1/81 1/243
Total length, N*R 1.33333 1.77777 2.37037 3.16049 4.21399
log N 0.60206 1.20412 1.80618 2.40824 3.01030
log (1/R) 0.47712 0.95424 1.43136 1.90848 2.38561
Dimsion
log N / (1/R)
1.26185 1.26185 1.26185 1.26185 1.26185

The dimension is always 1.26185 regardless of the iteration level! Hence,

D = log N/log(1/R) = 1.26185

Which can also be written, taking antilogarithms, as

N=(1/R)^1.26185

The formulas above indicate that N and R are related through a power law. In general , a power law is a (usually nonlinear) relationship that for our present purposes can be written in the form N=a(1/R)^D, where D is normally a non-integer constant and a is a numerical constant which in the case of the Koch curve is 1.