is likely to be very rugged itself. Just as with the broccoli, this self-similarity is (of course) not perfect: the San Francisco bay is not a perfect miniaturization of California’s coastline overall, but they look similar in many respects. Moreover, it turns out that the more rugged a coastline is, the higher fractal dimension it has: coasts with outlines that are very complex have higher fractal dimension than coasts that are relatively simple and smooth.
The most serious problem with using fractal dimension as a general measure of complexity is that it seems to chiefly be quantifying a fact about how complex an object’s spatial configuration is: the statistical self-similarity that both broccoli and coastlines show is a self-similarity of shape. This is just fine when what we’re interested in is the structure or composition of an object, but it isn’t at all clear how this notion might be expanded. After all, at least some of our judgments of complexity seem (at least at first glance) to have very little to do with shape: when I say (for instance) that the global economy is more complex today than it was 300 years ago, it doesn’t look like I’m making a claim about the shape of any particular object. Similarly, when I say that a human is more complex than a fern, I don’t seem to be claiming that the shape of the human body has a greater fractal dimension than the shape of a fern. In many (perhaps most) cases, we’re interested not in the shape of an object, but in how the object behaves over time; we’re concerned not with relatively static properties like fractal dimension, but with dynamical ones too. Just as with Shannon entropy, there seems to be a grain of truth buried in the fractal dimension measure, but it will takes some work to articulate what it is; also like Shannon entropy, it seems as though fractal dimension by itself will not be sufficient.
2.2 Moving Forward
We have spent the majority of this chapter introducing some of the concepts behind
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