sense of fractal dimension before considering the formal structure of mathematical fractals. Let’s begin by getting a handle on what counts as statistical self-similarity in nature, then, to begin with.
Consider a stalk of broccoli or cauliflower that we might find in the produce section of a supermarket. A medium-sized stalk of broccoli is composed of a long smooth stem (which may be truncated by the grocery store, but is usually still visible) and a number of lobes covered in what look like small green bristles. If we look closer, though, we’ll see that we can separate those lobes from one another and remove them. When we do, we’re left with several things that look very much like our original piece of broccoli, only miniaturized: each has a long smooth stem, and a number of smaller lobes that look like bristles. Breaking off one of these smaller lobes reveals another piece that looks much the same. Depending on the size and composition of the original stalk, this process can be iterated several times, until at last you’re removing an individual bristle from the end of a small stalk. Even here, though, the structure looks remarkably similar to that of the original piece: a single green lobe at the end of a long smooth stem.
This is a clear case of the kind of structure that generally gets called “fractal-like.” It’s worth highlighting two relevant features that the broccoli case illustrates nicely. First, fractal-like physical systems have interesting detail at many levels of magnification: as you methodically remove pieces from your broccoli stem, you continue to get pieces with detail that isn’t homogenous. Contrast this with what it looks like when you perform a similar dissection of (say) a carrot. After separating the leafy bit from the taproot, further divisions produce (no pun intended) pieces that are significantly less interesting: each piece ends up looking more-or-less
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