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Form my understanding, the Hilbert curve, for instance, can fill the unit square to arbitrary density through decomposition (replacing the current image with multiple scaled, rotated, and translated copies), or can fill an arbitrary amount of a 2-dimensional lattice through composition (duplication, rotation, and translation of the existing image without scaling or replacement).

If that's true, then the Hilbert curve should work as a pairing function, as any point on the lattice can be mapped to the number of points that must be traversed along the Hilbert curve to reach it.

Is that a correct understanding, and does it mean that any 2-dimensional space-filling curve can be a pairing function?

brianmearns
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    What do you mean by a "pairing function"? What do you mean by "the number of points that must be traversed ..."? Going along any arc of a curve you traverse infinitely many points! – Robert Israel Aug 02 '16 at 20:09
  • I think what you're asking can be stated more precisely as this: Is every space-filling curve a limit of some sequence of curves, each of which looks like a "rook's path" on some $n\times n$ grid? – Barry Cipra Aug 02 '16 at 20:30
  • By "pairing function" do you mean a bijection? – Gregory Grant Aug 02 '16 at 21:02
  • By pairing function, I mean a bijection from the set of all orders pairs of natural numbers to the natural numbers. With regards to the number of points traversed, in referring to lattice points visited. Or, if you prefer, the number of "unit length" line segments along the curve. – brianmearns Aug 02 '16 at 22:15

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You have a really detailed answer here

In short:

  • There is no continuous bijection from the unit segment to the unit square
  • There are bijections, and they are quite simple to make
rambi
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