Dimensions and Logarithms

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)L².

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

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,

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)L^D].

Now take the logarithm (log) of both sides:

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

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:

slope = $\displaystyle {\log N_2-\log N_1\over\log d_2-\log d_1}$ .

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