Complexity_ A Guided Tour - Melanie Mitchell [53]
Indeed, according to this definition of dimension, a line is one-dimensional, a square two-dimensional and a cube three-dimensional. All good and well.
Let’s apply an analogous definition to the object created by the Koch rule. At each level, the line segments of the object are three times smaller than before, and each level consists of four copies of the previous level. By our definition above, it must be true that 3dimension is equal to 4. What is the dimension? To figure it out, I’ll do a calculation out of your sight (but detailed in the notes), and attest that according to our formula, the dimension is approximately 1.26. That is, the Koch curve is neither one- nor two-dimensional, but in between. Amazingly enough, fractal dimensions are not integers. That’s what makes fractals so strange.
In short, the fractal dimension quantifies the number of copies of a self-similar object at each level of magnification of that object. Equivalently, fractal dimension quantifies how the total size (or area, or volume) of an object will change as the magnification level changes. For example, if you measure the total length of the Koch curve each time the rule is applied, you will find that each time the length has increased by 4/3. Only perfect fractals—those whose levels of magnification extend to infinity—have precise fractal dimension. For real-world finite fractal-like objects such as coastlines, we can measure only an approximate fractal dimension.
I have seen many attempts at intuitive descriptions of what fractal dimension means. For example, it has been said that fractal dimension represents the “roughness,” “ruggedness,” “jaggedness,” or “complicatedness” of an object; an object’s degree of “fragmentation”; and how “dense the structure” of the object is. As an example, compare the coastline of Ireland (figure 7.2) with that of South Africa (figure 7.4). The former has higher fractal dimension than the latter.
One description I like a lot is the rather poetic notion that fractal dimension “quantifies the cascade of detail” in an object. That is, it quantifies how much detail you see at all scales as you dive deeper and deeper into the infinite cascade of self-similarity. For structures that aren’t fractals, such as a smooth round marble, if you keep looking at the structure with increasing magnification, eventually there is a level with no interesting details. Fractals, on the other hand, have interesting details at all levels, and fractal dimension in some sense quantifies how interesting that detail is as a function of how much magnification you have to do at each level to see it.
This is why people have been attracted to fractal dimension as a way of measuring complexity, and many scientists have applied this measure to real-world phenomena. However, ruggedness or cascades of detail are far from the only kind of complexity we would like to measure.
FIGURE 7.4. Coastline of South Africa. (Photograph from NASA Visible Earth [http://visibleearth.nasa.gov].)
Complexity as Degree of Hierarchy
In Herbert Simon’s famous 1962 paper “The Architecture of Complexity” Simon proposed that the complexity of a system can be characterized in terms of its degree of hierarchy: “the complex system being composed of subsystems that, in turn, have their own subsystems, and so on.” Simon was a distinguished political scientist, economist, and psychologist (among other things); in short, a brilliant polymath who probably deserves a chapter of his own in this book.
Simon proposed that the most important common attributes of complex systems are hierarchy and near-decomposibility. Simon lists a number of complex systems that are structured hierarchically—e.g., the body is composed of organs, which are in turn composed of cells, which are in turn composed of celluar subsystems, and so on. In a way, this notion is similar to fractals in the idea that there are self-similar patterns at all scales.
Near-decomposibility refers to the fact that, in hierarchical complex systems,