Is any space-filling curve a pairing function?

Question

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?

Answer 1

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