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Fig. 2.1

Look familiar? This shape[1] is starting to look suspiciously like our stalk of broccoli: there’s a main “stem” formed by the first few shapes (and the negative space of later shapes), “lobes” branching off from the main stem with stems of their own, and so on. If you could iterate this procedure an infinite number of times, in fact, you’d produce a perfect fractal: you could zoom in on almost any region of the shape and find perfect miniaturized copies of what you started with. Zooming in again on any region of one of those copies would yield even more copies, ad infinitum.

This is a neat mathematical trick, but (you might wonder) what’s the point of this discussion? How does this bear on complexity? Stay with me just a bit longer here—we’re almost there. To explain the supposed connection between fractal-like systems and complexity, we have to look a bit more closely at some of the mathematics behind geometric fractals; in particular, we’ll have to introduce a concept called fractal dimension. All the details certainly aren’t necessary for what we’re doing here, but a rough grasp of the concepts will be helpful for what follows. Consider, to begin with, the intuitive notion of “dimension” that’s taught in high school math classes: the dimensionality of a space is just a specification of how many numbers need to be

  1. The shape generated by this procedure is called the Pythagoras Tree.