same sense that nothing in nature is actually a circle. In the case of circles, we know exactly what it means to say that there are no circles in nature: no natural systems exist which are precisely isomorphic to the equation that describes a geometric circle: things (e.g. basketball hoops) are circular, but on close enough examination they turn out to be rough and bumpy in a way that a mathematical circle is not. The same is true of fractals; if we continue to subdivide the broccoli stalk discussed above, eventually we’ll reach a point where the self-similarity breaks down—we can’t carry on getting smaller and smaller smooth green stems and round green bristles forever. Moreover, the kind of similarity that we see at each level of magnification is only approximate: each of the lobes looks a lot like the original piece of broccoli, but the resemblance isn’t perfect—it’s just pretty close. That’s the sense in which fractal-like physical systems are only statistically self-similar—at each level of magnification, you’re likely to end up with a piece that looks more-or-less the same as the original one, but the similarity isn’t perfect. The tiny bristle isn’t just a broccoli stalk that’s been shrunk to a tiny size, but it’s almost that. This isn’t the case for mathematical fractals: a true fractal has the two features outlined above at every level of magnification—there’s always more interesting detail to see, and the interesting details are always perfectly self-similar miniature copies of the original
Here’s an example of an algorithm that will produce a true fractal:
- Draw a square.
- Draw a 45-45-90 triangle on top of the square, so that the top edge of the square and the base of the triangle are the same line. Put the 90 degree angle at the vertex of the triangle, opposite the base
- Use each of the other two sides of the triangle as sides for two new (smaller) squares.
- Repeat steps 1-4 for each of the new squares you’ve drawn.
Here’s what this algorithm produces after just a dozen iterations: