Free Simply Complexity by Neil Johnson Page A

right-hand side of figure 3.2 , is between being an infinite set of points and a solid line. Now, it turns out that scientists refer to a point as being zero-dimensional, a line as being one-dimensional, and a flat sheet such as a television screen as being two-dimensional. This fine dust of points which looks like a solid line but isn’t, is effectively between a point and a line – hence it is between a zero-dimensional object and a one-dimensional object. As we all know, a number between zero and one is called a fraction – hence the fine dust of points has a fractional dimension. For this reason, scientists call an object such as this fine dust of points, a fractal.
    You can also see a repeated pitchfork shape emerge as the value of
r
increases toward 3.6. Each line splits into two – and this repeats itself on an ever smaller scale. It is equivalent to saying that the period keeps doubling from 2 to 4 to 8 etc. As we noted above, the final object is a line of points – like a dust of points – which would look to your eye as you move toward the page as “points within points within points”. Here we have the same thing: a pitchfork-like pattern within a pitchfork-like pattern within a pitchfork-like pattern. This pattern-within-a-pattern repeated over and over is again referred to as a fractal.
    This emergence of fractals is a common occurrence in Complex Systems, both in terms of the output which a Complex System produces in time and the resulting shapes which emerge in space. In other words, fractals are a typical emergent phenomenon of Complex Systems. Just as with Chaos, this does not mean that fractals are always observed in a Complex System – just that they can be. Given this widespread interest in fractals, the two boxes marked “Fractal Fun” show a couple of ways in which you can generate fractals using just a pen and paper. This is not how real-world Complex Systems actually generate their fractals – far from it, since there are many ways of generating the same fractal. But they do help illustrate what a fractal is.
    Suppose that our systematic intern is now dictating the price of some commodity in a market, as opposed to dictating the location of a file. The position of the file
S
becomes the market price.Instead of a time-series of file locations as we had above, we would have a series of prices – a price time-series. The rich variety of behavior that this price series could then show is indeed consistent with the wide range of behaviors that we see emerging from financial markets – from moments where the price doesn’t seem to change (e.g.
r
= 0.1), to moments where it appears to oscillate back and forth in a so-called business cycle (e.g.
r
= 3.2), through to moments where it appears to be random (e.g.
r
= 4). In other words if our intern-turned-price-maker increased the value of
r
from zero to four, the price would range in behavior from something that never changed over time, to something that repeated itself, to something that looked quite simply chaotic.
    Fractal Fun I: Dust to dust
     

     
    The figure above shows how to produce a dust-like fractal. Start by drawing a horizontal line. Divide it into three, and remove the middle piece. This leaves you with a straight line with a hole in the middle. In fact the easiest way to do this, is to draw the new shape you generate in an empty space below the present shape, as shown. Treating each of the two resulting pieces as a new line, divide each one into three and again remove the middle piece. Keep repeating this process over and over again, for as long as you can. You will end up with a fine dust of points – in other words, a fractal having a fractal dimension between zero and one. This fractal looks very similar to the one that turns up in the regime of Chaos in figure 3.2 .
     
    Fractal Fun II: Let it snow, let it snow, let it snow
     

     
    The figure above shows how to produce a snowflake-like fractal. Start by again drawing a