On 2015-12-23 12:00, Warren D Smith wrote:
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).
Sequences of curves are beguiling and misleading, by concealing that something singular happens as D reaches 2. The textures and designs that seem to distinguish these curves are just superficial treatments applied to the bottommost recursion. Recall that traditional pictures of Heighway's Dragon show whole grids of self-contacts. But simply joining the midpoints of its segments produces a completely self-avoiding "median curve": gosper.org/twindrag.png . Alternatively, sampling the limiting "curve" at a frequency slightly off from a power of 2 produces beat frequencies: gosper.org/insuck.png showing whole intervals of phases where the polygonal approximations self-avoid. Finally, recall the plain old square grid four-around-one (base i+2, digits 0, i^(0..3)): gosper.org/erez2.png and how different it looks when quarter-circles replace the line segments at the bottom level: gosper.org/erez7.PNG Bottom line: There is no bottom level. --rwg An interesting rendition of a spacefill might be to use Julian's inverter on a big bunch of double and triple points, darkening each one in proportion to the separation of its inverse images. I expect it would resemble leaf veins.