original discussion of fractals, offers an alternative definition: a fractal is a figure where the fractal dimension is greater than the topological dimension.
At last, we’re in a position, then, to say what it is about fractals that’s supposed to capture our notion of complexity. Since fractal dimension quantifies the relationship between the proliferation of detail and the change in magnification scale, an object with a higher fractal dimension will show more interesting detail than an object with a lower fractal dimension, given the same amount of magnification. In the case of objects that are appropriately called “fractal-like” (e.g. our stalk of broccoli), this cascade of detail is more significant than you’d expect it to be for an object with the sort of abstract (i.e. topological) structure it has. That’s what it means to say that fractal dimension exceeds topological dimension for most fractals (and fractal-like objects): the buildup of interesting details in a sense “outruns” the buildup of other geometric characteristics. Objects with higher fractal dimension are, in a sense, richer and more rewarding: it takes less magnification to see more detail, and the detail you can see is more intricately structured.
So is this measure sufficient, then? You can probably guess by now that the answer is ‘no, not entirely.’ There are certainly cases where fractal dimension accords very nicely with what we mean by ‘complex:’ it excels, for instance, at tracking the rugged complexity of coastlines. Coasts—which were among Mandelbrot’s original paradigm cases of fractal-like objects—are statistically self-similar in much the same way that broccoli is. Viewed from high above, coastlines look jagged and irregular. As you zoom in on a particular section of the coast, this kind of jaggedness persists: a small segment of shore along a coast that is very rugged in general
- Mandelbrot offered these two definitions as equivalent. It has since been discovered, though, that there are a number of fractals (in the first sense) for which the latter definition does not hold. See Kraft (1995) for more on this.