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
The constant
depends on the shape. As examples, Figure shows shapes with three
different values of this constant.
[
画像:1.png]
Figure
2.10: The area of three 2-dimensional figures, showing three values
of the constant in the area formula (Area) = (Constant)L2.
Now we cover any of these shapes with little boxes of width d and area
d2. How many boxes N does it take? The following formula is
approximately correct:
(Area) = Nd2 = (Const)L2 = (Const)$\displaystyle \left(\vphantom{L{d\over d}}\right.$L$\displaystyle {d\over d}$$\displaystyle \left.\vphantom{L{d\over d}}\right)^{2}_{}$ = (Const)$\displaystyle \left(\vphantom{{L\over d}}\right.$$\displaystyle {L\over d}$$\displaystyle \left.\vphantom{{L\over d}}\right)^{2}_{}$d2,
or
Nd2 = (Const)$\displaystyle \left(\vphantom{{L\over d}}\right.$$\displaystyle {L\over d}$$\displaystyle \left.\vphantom{{L\over d}}\right)^{2}_{}$d2.
Cancel the factor
d2 on both sides of the equation to obtain:
N = (Const)$\displaystyle \left(\vphantom{{L\over d}}\right.$$\displaystyle {L\over d}$$\displaystyle \left.\vphantom{{L\over d}}\right)^{2}_{}$ = d-2[(Const)L2].
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:
N = d-D[(Const)LD].
Now take the logarithm (log) of both sides:
log N = log(d-D) + log[(Const)LD].
Here
L is fixed; we are not changing the area or the overall width
L
of the figure as we use boxes of different width
d to cover
it. Therefore everything in the square bracket is a constant, and the
log of the quantity in the square bracket is also a constant, which we
can call ``Constant'' Using the property of logs, we have:
log N = - D log d + Constant.
Think of the variables as log
N and log
d rather than
N and
d. Then this can be thought of as the equation of a straight line with
slope -
D. Hence our
log-log box-covering plot will yield a
straight line whose slope is the negative of the dimension
D.
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:
