
Nonlinear Geoscience Fractals 

Determining Fractal DimensionWe 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 selfsimilar 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 selfsimilar. 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 selfsimilar 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 twodimensional and a cube is threedimensional. 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!
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 noninteger constant and a is a numerical constant which in the case of the Koch curve is 1. 