Page:Lawhead columbia 0054D 12326.pdf/78

 the same as the last one—smooth, orange, and fibrous. That’s one feature that makes fractal-like parts of the world interesting, but it’s not the only one. After all, it’s certainly the case that there are many other systems which, on dissection, can be split into pieces with interesting detail many times over—any sufficiently inhomogeneous mixture will have this feature. What else, then, is the case of fractals tracking? What’s the difference between broccoli and (say) a very inhomogeneous mixture of cake ingredients?

The fact that (to put it one more way) a stalk of broccoli continues to evince interesting details at several levels of magnification cannot be all that makes it fractal-like, so what’s the second feature? Recall that the kind of detail that our repeated broccoli division produced was of a very particular kind—one that kept more-or-less the same structure with every division. Each time we zoomed in on a smaller piece of our original stalk, we found a piece with a long smooth stem and a round green bristle on the end. That is, each division (and magnification) yielded a structure that not only resembled the structure which resulted from the previous division, but also the structure that we started with. The interesting detail at each level was structurally similar to the interesting detail at the level above and below it. This is what separates fractal-like systems from merely inhomogeneous mixtures—not only is interesting detail present with each division, but it looks the same. Fractal-like systems (or, at least the fractal-like systems we’re interested in here) show interesting details at multiple levels of magnification, and the interesting details present at each level are self-similar.

With this intuitive picture on the table, let’s spend a moment looking at the more formal definition of fractals given in mathematics. Notice that we’ve been calling physical systems “fractal-like” all along here—that’s because nothing in nature is actually a fractal, in just the 68