J?rg, these recursive arcs are beautiful and ingenious, but in the limit, they all describe the same area-filling function, where in this case, the area filled is 1/3 of a "France Flake" island. All such area-fills map closed intervals onto closed sets, hitting *all* the points at least once, uncountably many at least twice, and at least countably many at least thrice.
--it seems to me, there must be some value in an area-filling curve that is the "limit" of a sequence curve1, curve2, ..., in which every curveN never self-intersects and "stays away from intersecting itself" by at least some natural distance f(N). Since in any real application, N will be finite. Possible way to make that precise: Each point of curve N is distance >= f(N) away from any other point of curveN that is not at ArcDistance <= 2*f(N). So then the question is: which area-filling curves can be manufactured in this way, and which cannot? One might argue they all can be done (proof: local surgery as needed) but perhaps not in a "nice" way (simple nice definition).