It won't surprise you to learn that these aren't "curves" in the sense that you might think of them; if I ask you to draw a "curve" you'll probably draw something that's what mathematicians would call "piecewise smooth". What this means, roughly, is that you can draw a piece of it without having any "kinks", then turn, then draw another such piece, and so on, doing this only a finite number of times. Space-filling curves don't have this property; they are made up of infinitely many such "pieces". Not surprisingly, they also have infinite length. These curves are made by an iterative process; in the case of the Hilbert curve:
- on the first iteration the curve has length 3/2 and each point is within √2/4 of the curve;
- on the second iteration the curve has length 15/4 and each point is within √2/8 of the curve;
- on the third iteration the curve has length 63/8 and each point is within √2/16 of the curve;
- on iteration n the curve has length 2n - 1/2n (it is made of 4n-1 segments of length 2-n) and each point is within √2/2n+1 of the curve.
The maximum distance halves and the length doubles with each step; as we iterate, each point in the square is arbitrarily close to a point on the curve (thus on the curve) and the curve is infinitely long.
Andrew Cook at "Statistical Modeling, Causal Inference, and Social Science" writes about the fractal nature of scientific revolutions, pointing to this earlier post of his. The idea is that science moves forward in what the evolutionary biologists call "punctuated equilibrium" -- at most points "not much" is getting done but occasionally big moves are made and in the end science gets done. (This is a bit unfair, though, because the scientists who are doing the "not much" are often collecting the sort of data that is exactly what the revolutinaries doing the paradigm shift will turn out to need.) If this is true, then we might say that all the science that will get done between year 0 ("now") and year 81 (which turns out to be 2088) gets done either in the first third of that period (between 0 and 27) or the last third (between 54 and 81). But then something similar happens on each of those periods -- all the science gets done between 0 and 9, 18 and 27, 54 and 63, or 80 and 81. If we repeat this, ad infinitium, we get that the set of times at which science is being done is the Cantor set, which has measure zero; furthermore the rate of scientific progress, when scientific progress is happening, must be infinite in order for any science to happen at all!
Of course, this is ridiculous. But it makes sense that science happens in bursts, and that each burst is made of smaller bursts, and so on; that there are periods of stasis between these bursts, but that some of these periods of stasis are more static than others; and so on. It's only the mathematician's insistence on taking the limit that makes this model not work. Furthermore, there's more than one kind of science, and it could happen that one discipline's burst is another discipline's period of stasis. And maybe a model like this is more likely to hold for the individual scientist (who has periods when they Get Things Done and periods when they don't) than for science as a whole.
But the periods when it looks like the scientist isn't doing anything might be essential. The subconsious is often doing work then. Perhaps there is something about the way our subconscious works -- in which bigger breakthroughs need longer fallow periods to precede them -- that leads to this fractal nature, with bursts upon bursts.