Fractal geometry is the study of notions of "dimension" that might take on fractional values. The simplest such notion of dimension is called similarity dimension and is applicable to sets that are strictly self-similar.
A set is called self-similar if it is composed of scaled copies of itself. The Koch curve shown below, for example, consists of four copies of itself scaled by the factor \(1/3\).
The Koch curve is crinkly - very crinkly. The self-similarity of the Koch curve implies that the crinkles you see at the macro-level appear everywhere, no matter how far you zoom in. This property will be reflected in its "dimension", which should be at least one but less than two. Somehow, it's complicated enough that its dimension might be between one and two.
A self-similar set doesn't have to be complicated, though. In fact, there are some very important self-similar sets that are quite simple. The most basic such examples are the line segment, the square, and the cube. These can all be broken into smaller copies of themselves in a number of ways.
| line segment | square | cube | |
| \(r=1/2\) | \(N=2\) | \(N=4=2^2\) | \(N=8=2^3\) |
| \(r=1/3\) | \(N=3\) | \(N=9=3^2\) | \(N=27=3^3\) |
| \(r=1/5\) | \(N=5\) | \(N=25=5^2\) | \(N=125=5^3\) |
In the simplest situation, all the copies are scaled by the same factor let's say \(r\). A glance at the table above shows that (for those sets, at least), the scaling factor \(r\), the number of pieces \(N\), and the dimension of the set \(d\) satisfy a simple formula:
\[N = (1/r)^d.\]This suggests that we solve for \(d\) and define the similarity dimension of a self-similar set to be
\[d = \frac{\log(N)}{\log(1/r)}.\]This definition automatically assigns the correct notion of dimension to the simple sets that motivated the defintion. However, it's also applicable to sets like the Koch curve, which is assigned dimension
\[d = \frac{\log(4)}{\log(3)} \approx 1.26186.\]Some sets are self-similar but with more than one scaling factor in any decomposition. The curve below, for example, is called the z-curve. It consists of three copies of itself scaled by the factor \(1/3\) and two copies of itself scaled by the factor \(\sqrt{2}/6\).
In general, we might have a list of scaling factors \(r_1,r_2,\ldots,r_m\). The dimension is then the unique number \(d\) with the property that
\[r_1^d + r_2^d + \cdots +r_m^d = 1.\]Justification of this is a bit beyond what we're doing in this class, but I will point out that, all the scaling factors are the same, then this boils down to our simpler formula with the logarithms. This is because, if \[r_1 = r_2 = \cdots = r_m = r,\] then \[r_1^d + r_2^d + \cdots +r_m^d = mr^d = 1\] or \(m=1/r^d\), which is exactly our first equation for similarity dimension.
As an example, the dimension of the z-curve is the unique number \(d\) such that
\[3(1/3)^d + 2(\sqrt{2}/6)^d = 1.\]The solution to this equation is approximately, \(d\approx1.32038\), as Wolfram|Alpha will tell you.
There's a nice discussion on math.stackexchange concerning the solution of these types of equations.