2.4 - Dimensions and Logarithms

Let's
derive the equations that justify the measurement of the dimension
of an object as the magnitude of the slope of a straight line on
a log-log graph. Think of a pattern that has a fixed area and fixed
overall width L. We are going to cover this pattern with square
boxes of width d and count the number N of the boxes needed to cover
it. For a solid area, we have a dimension of 2 and the general formula

(Area) = (Constant)*L*^{2}.

The constant
depends on the shape. As examples, Figure shows shapes with three
different values of this constant.

Now we cover any of these shapes with little boxes of width *d* and area
*d*^{2}. How many boxes *N* does it take? The following formula is
approximately correct:

(Area) = *Nd*^{2} = (Const)*L*^{2} = (Const)*L* = (Const)*d*^{2},

or
This result is for a 2-dimensional object, such as those shown in Figure
2.10. For a fractal, the dimension is not necessarily 2. Call
the dimension *D*. Then the corresponding equation becomes:

log *N* = log(*d*^{-D}) + log[(Const)*L*^{D}].

Here
log *N* = - *D* log *d* + Constant.

Think of the variables as log
Take care when measuring the slope *D* to not use the numbers
along the logarithmic scales. Instead, measure this slope directly, that
is, with an ordinary ruler, as shown in
Figure 2.3, or use the formula:

slope = .