# Complex Systems, Emergence, and Multiscale Analysis: A Tutorial and Brief Survey
## 2 Basics of Complex Systems and Emergence
### Basics of Random Fractal Theory
#### 2.4.1 Introduction to Fractal Theory
Based on this simple calculation, we now understand when we walk along a wild trail, get fired, slow down, we are actually shrinking our step sizes, so we will be walking out a longer trail! Next, we consider the Cantor set, one of the prototypical fractal objects, so that we can appreciate the concept of fractal dimension better. The standard Cantor set is obtained by first partitioning a line segment into three equal parts and deleting the middle one. This step is then repeated, deleting the middle third of each remaining segment iteratively. See Figure 1(a). Note that such a process can be related to the iteration of a nonlinear discrete map. The removed middle thirds can be related to the intervals that make the map diverge to infinity, while the remaining structures are linked to the invariant points of the map. At the limiting stage, \(n\rightarrow\infty\), the Cantor set consists of infinitely many isolated points. Consistent with isolated point(s) having dimension 0, the topological dimension here is 0. The length of the total segments removed is  
\[\frac{1}{3}+2\times\left(\frac{1}{3}\right)^{2}+4\times\left(\frac{1}{3}\right) ^{3}+\cdots=\frac{1}{2}\times\frac{2}{3}+\frac{1}{2}\times\left(\frac{2}{3} \right)^{2}+\frac{1}{2}\times\left(\frac{2}{3}\right)^{3}+\cdots=1 \tag{69}\]  
Therefore, the entire unit interval has been removed! Is the fractal dimension here the same as the topological dimension, which is 0? To find out, let us focus on stage \(n\). One needs \(N(\epsilon)=2^{n}\) boxes of length \(\epsilon=(\frac{1}{3})^{n}\) to cover the set. Hence, the fractal dimension for the Cantor set is  
\[D=-\ln N(\epsilon)/\ln\epsilon=\ln 2/\ln 3. \tag{70}\]  
It is not zero!The fractal dimension of the Cantor set can also be computed by employing the self-similar feature. Denote the number of intervals needed to cover the Cantor set at a certain stage with scale \(\epsilon\) by \(N(\epsilon)\). When the scale is reduced by 3, \(N(\epsilon/3)\) is doubled. Since \(N(\epsilon/3)/N(\epsilon)=3^{D}=2\), one immediately gets \(D=\ln 2/\ln 3\). To better transit to random fractals, we note that the standard Cantor set can be made random. One way is to divide each interval into three equal parts and randomly delete one of them (see Figure 20). An alternative way of obtaining a random Cantor set is to delete a middle interval of random length at each stage (Figure 20). Clearly, case (b) also has dimension \(D=\ln 2/\ln 3\). When certain regulation is imposed on the length distribution for the subintervals in case (c), the fractal dimension can also be readily computed. One way of imposing such a regulation is to require that the ratio of the subinterval and its immediate parent interval follows some distribution that is stage-independent. Such a regulation is essentially a multifractal construction, which we will discuss soon. The above discussion suggests that two different geometrical fractals may have the same fractal dimension. To further appreciate this point, we have shown in Figure 20, a different type of regular Cantor set. It is obtained by retaining four equally spaced segments whose length is 1/9 th of the preceding segment. Denote the number of segments at a certain stage with length scale \(\epsilon\) by \(N(\epsilon)\). When the scale is reduced by 9, \(N(\epsilon/9)\) is quadrupled. Here, \(D\) is again \(\ln 2/\ln 3\), since \(N(\epsilon/9)/N(\epsilon)=9^{\text{D}}=4\). Based on the above discussions, one can readily realize that fractal curves and surfaces are more space filling. This property is beneficial in biological evolution. As a result, fractal forms are abundant in biology. Instead of giving actual examples here, we will refer readers to reference [122] for a menagerie of fractal forms in living things. This more space-filling property of fractals has also been exploited to design fractal antennas by maximizing the effective length or perimeter of the material that receives or transmits electromagnetic radiation. Fractal antennas are excellent for wideband and multiband applications [123].