Free Simply Complexity by Neil Johnson Page B

horizontal line. Divide it into three, but now replace the middle piece by two pieces which are equal in length to the removed piece. This leaves you with a straight line with a hat shape in the middle, as shown. Treating each of the resulting pieces as a new line, divide each one into three and again replace the middle piece by two pieces of equal length. Keep repeating this over and over again. This time the fractal looks like it contains so may lines that it begins to fill the page. In other words, this snowflake-like structure has a fractal dimension between one and two. Similar shapes can arise in many real-world systems – for example, the border of a cancer tumor.
     
    Now, if all this can happen with such a simple setup as one systematic intern and a filing cabinet, imagine what could happen in a system containing a collection of such people or objects? The short answer is “all this and more”. But don’t worry – we haven’t been wasting our time. It turns out that the range of behaviors which arise in this simple example tend to be ones that are often observed in real-world Complex Systems. For example, the heart is a Complex System containing a collection of cells which are interacting in a complicated way, with feedback. The result is an output which seems to oscillate, or “beat”, in a fairly ordered way – but it can occasionally behave erratically. Meanwhile a market price is typically random looking – but it can occasionally appear more regular with oscillatory-type behavior emerging every so often.
3.4 If I remember correctly, I am Iiving on the edge
     
    We just saw how a systematic intern could generate Chaos, and hence fractals, in the location of a file in a filing cabinet. I also gave you two mathematical rules which had nothing to do with files or interns, but which still managed to generate fractals. Now let’s have a look at those fractals again. One could argue that the dust-like fractal shown in figure 3.2 and in “Fractal Fun I”, looks a bit like traffic seen from way up in the sky – or even a trail of ants. And the snowflake-like fractal in “Fractal Fun II” could be said tolook like the coastline of an island. Its jaggedness might also make you think a bit about mountain ranges, or even those stock price-charts that we mentioned. Because of this apparent similarity, it is precisely at this stage of the discussion of Complexity that many articles and books on the subject tend to stop dead. Here is their reasoning: a systematic rule can produce Chaos; Chaos has fractals in it; fractals look a bit like things we see all around us; so they must all be one and the same thing. Case closed.
    But that kind of reasoning does very rough justice to the whole Complexity mystery. Indeed just like an investigation into a crime, where finding one possible motive doesn’t mean we have found
the
motive, finding one possible way of generating complex emergent behavior doesn’t mean we have found
the
way it is actually generated in real-world Complex Systems. Just take the dust-like fractal, and think of traffic. Fractal patterns are indeed observed occasionally in everyday traffic – however there is no systematic intern organizing the traffic and thereby producing the observed fractal pattern. Nor is the line of drivers on the road magically dividing itself into three, removing the middle section, and hence going through the fractal-generating rule I gave you in the previous section. Even though it is encouraging to have shown that relatively simple rules can give such phenomena, it in no way explains how a Complex System comprising many interacting parts, manages to produce such phenomena. This is precisely the reason why this book won’t go the route that so many others have by focusing on different types of mathematical rules for producing non-linear dynamical phenomena such as Chaos and fractals. There is already a branch of science that does that, and people like Steve Strogatz have