# 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  
Euclidean geometry studies simple shapes, including lines, planes, triangles, squares, cones, spheres, and so on. All these shapes are regular. Every one has seen clouds, mountains, and other complex shapes in nature. How well can those complex shapes be modeled by circles, spheres, cones, or other regular shapes? Very badly! When thinking along this line, [PERSON] has created a new field, the fractal geometry [121]. Let us first try to understand fractal intuitively. The key here is self-similarity, which means that part of an object, when magnified, is similar to the whole. More concretely, whether we magnify the object by 10 times or 100 times, we always observe similar objects. When discussing power laws, we have emphasized that a power law embodies self-similarity (please see Figure 2). Therefore, power law relations are natural mathematical tools to characterize fractals. When plotted in double-logarithmic scale, power laws become linear relations. To better appreciate the significance of power laws, imagine hiking on a mountain trail. Unlike many manmade trails with hundreds of stairs in the mountains of China, we assume the trail we are walking up is wild and irregular. How can we measure the distance we have walked? Let us measure the total distance by our step size. Denote it by \(\epsilon\). Note \(\epsilon\) could be different for different hikers--one who rides a horse has a huge step size, while a little baby surely only has a tiny step size. The distance we have walked up is then  
\[L=N(\epsilon)\cdot\epsilon, \tag{66}\]  
where \(N(\epsilon)\) is the number of intervals walked. Amazingly, \(N(\epsilon)\) scales with \(\epsilon\) as a power law, just as described by Equation (23), where \(1<D<2\) is not an integer. Such a nonintegral \(D\) is the celebrated fractal dimension of the hiking trail. What is the meaning of a nonintegral \(D\)? To find the answer, we start from the measurement of certain length, area, or volume. The basics of calculus teach us that we can measure a curve, a surface area, or a volume using very small intervals, squares, or cubes by properly covering the object we are interested to measure. Take the unit length, unit area, or unit volume as the unit of measurement, with linear size \(\epsilon\). Now suppose we measure the length of a straight line with length 1. What is the minimal number of intervals of length \(\epsilon\) needed to fully cover this unit length? We need at least \(N(\epsilon)\sim\epsilon^{-1}\) intervals. Extending to 2D and 3D, when covering an area or volume by boxes  
becomes \(N(\epsilon)=Me^{-D}\), with \(D=0\). Therefore, the topological dimension \(D\) for isolated points is zero. We thus see that when we call a point, a line, an area, and a volume \(0-D\), \(1-D\), \(2-D\), and \(3-D\) objects, we are talking about their topological dimensions. We can now discuss the consequence of \(1<D<2\) for an irregular mountain trail. Combining Equations (23) and (66), we have  
\[L=\epsilon^{1-D}, \tag{67}\]  
Therefore, when \(\epsilon\) becomes smaller, \(L\) becomes larger. In fact, when \(\epsilon\to 0\), \(L\rightarrow\infty\). This property was actually first found by [PERSON], a mathematician, meterologist, and pacifist who devoted himself in his later years to the study of the causes of wars and ways to prevent them. However, we have to wait for [PERSON] to find the quantitative power law relation described by Equation (23), to create the new field of fractal. With Equation (67), we can actually deduce more by using some concrete numbers. For example, let us take \(D=1.25\), and imagine a race between a hare and a tortoise. Taking into account the physical difference between a hare and a tortoise, it it reasonable to assume that the step size of the hare is 16 times that of the tortoise. Then we have  
\[L_{\mbox{hare}}=\frac{1}{2}L_{\mbox{tortoise}} \tag{68}\]  
The tortoise has to crawl twice the distance that the hare runs!