Fractals are unfair

Let's say you want to identify an arbitrary point in a segment I (chosen with an uniform probability distribution). A more or less canonical way to do this is to split the segment in two equal halves, and write down a bit identifying which half; now the size of the set where the point is hidden is one half of what it was. Because the half-segment we chose is affinely equivalent to the original one, we can repeat this as much as we want, gaining one bit of precision (halving the size of the "it's somewhere around here" set) for each bit of description. Seems fair.

It's easy to do the same on a square I in R^2. Split the square in four squares, write down two bits to identify the one where the point is, repeat at will. Because each square has one fourth the size of the enclosing one, you gain two bits of precision for each two bits of description. Still fair (and we cannot do better).

Now try to do this on a fractal, say Koch's curve, and things go as weird as you'd think. You can always split it in four affinely equivalent pieces, but each of them is one-third the size of the original one, which means that you gain less than two bits of precision for each two bits of description. Now this is unfair. A fractal street would be a very good way of putting in an infinite number of houses inside a finite downtown, but (even approximate) driving directions will be longer than you'd think they should.

Of course, this is merely a paraphrasing of the usual definition of a fractal, which is an object whose fractal dimension exceeds its topological dimension (very hand-wavingly speaking, the number of bits of description in each step is higher than the number of bits of precision that you gain). But I do enjoy looking at things through an information theory lens.

Besides, there's a (still handwavy but clear) connection here with chaos, through the idea of trying to pin down the future of a system in phase space by pinning down its present. In conservative systems this is fair: one bit of precision in the present, gives you one bit of precision for the future (after all, volumes are preserved). But when chaos is involved this is no longer the case! For any fixed horizon, you need to put in a more bits of information about the present in order to get the same number of bits about the future.