Section 5.1 Quantifying dimension
Both of the definitions of dimension that we'll consider (similarity dimension and box-couting dimension) are generalizations of a very simple idea. We attempt to quantify the dimension of some very simple sets in a way that generalizes to more complicated sets. The simple sets we consider are the unit interval \([0,1]\text{,}\) the unit square \([0,1]^2\text{,}\) and the unit cube \([0,1]^3\text{,}\) which should clearly have dimensions 1, 2, and 3 respectively. Each of these can be decomposed into some number \(N_r\) of copies of itself when scaled by certain factors \(r\text{.}\) The following table shows the values of \(N_r\) for various choices of \(r\) and for each of these simple objects.
| \([0,1]\) | \([0,1]^2\) | \([0,1]^3\) | |
| \(r=1/2\) | \(N_r=2=2^1\) | \(N_r=4=2^2\) | \(N_r=8=2^3\) |
| \(r=1/3\) | \(N_r=3=3^1\) | \(N_r=9=3^2\) | \(N_r=27=3^3\) |
| \(r=1/5\) | \(N_r=5=5^1\) | \(N_r=25=3^2\) | \(N_r=125=5^3\) |
The decomposition is illustrated for \(r=1/3\) in Figure 5.1.2
Note that no matter the scaling factor or set, the number of pieces \(N_r\text{,}\) the scaling factor \(r\text{,}\) and the dimension \(d\) are related by \(N_r = (1/r)^d\text{.}\) This motivates the following definition: If \(E\subset \mathbb{R}^n\) can be decomposed into \(N_r\) copies of itself scaled by the factor \(r\text{,}\) then
Many sets constructed via iterated function systems can be analyzed via equation (5.1.1). For example, the Cantor set is composed of two copies of itself scaled by the factor \(1/3.\) Thus it’s fractal dimension is \(\log(2)/\log(3).\) The fractal dimension of the Sierpinski gasket is \(\log(3)/\log(2)\) and the fractal dimension of the Koch curve is \(\log(4)/\log(3)\text{.}\)