Editing Space-filling curve (section)

== Properties ==
[[File:ComparingSFCurves-MortonHilbert1024.png|thumb|520px|[[Z-order curve|Morton]] and [[Hilbert curve|Hilbert]] curves of level 6 (4<sup>5</sup>=1024 cells in the [[Recursion (computer science)|recursive square partition]]) plotting each address as different color in the [[RGB color model|RGB standard]], and using [[Geohash]] labels. The neighborhoods have similar colors, but each curve offers different pattern of grouping similars in smaller scales.]]

If a curve is not injective, then one can find two intersecting ''subcurves'' of the curve, each obtained by considering the images of two disjoint segments from the curve's domain (the unit line segment).  The two subcurves intersect if the [[intersection (set theory)|intersection]] of the two images is [[Empty set|non-empty]].  One might be tempted to think that the meaning of ''curves intersecting'' is that they necessarily cross each other, like the intersection point of two non-parallel lines, from one side to the other.  However, two curves (or two subcurves of one curve) may contact one another without crossing, as, for example, a line tangent to a circle does.

A non-self-intersecting continuous curve cannot fill the unit square because that will make the curve a [[homeomorphism]] from the unit interval onto the unit square (any continuous [[bijection]] from a [[compact space]] onto a [[Hausdorff space]] is a homeomorphism).  But a unit square has no [[cut-point]], and so cannot be homeomorphic to the unit interval, in which all points except the endpoints are cut-points. There exist non-self-intersecting curves of nonzero area, the [[Osgood curve]]s, but by [[Netto's theorem]] they are not space-filling.{{sfn|Sagan|1994|p=131}}

For the classic Peano and Hilbert space-filling curves, where two subcurves intersect (in the technical sense), there is self-contact without self-crossing. A space-filling curve can be (everywhere) self-crossing if its approximation curves are self-crossing.  A space-filling curve's approximations can be self-avoiding, as the figures above illustrate.  In 3 dimensions, self-avoiding approximation curves can even contain [[knot theory|knots]].  Approximation curves remain within a bounded portion of ''n''-dimensional space, but their lengths increase without bound.

Space-filling curves are special cases of [[fractal curves]].  No differentiable space-filling curve can exist. Roughly speaking, differentiability puts a bound on how fast the curve can turn. Michał Morayne proved that the [[continuum hypothesis]] is equivalent to the existence of a Peano curve such that at each point of the real line at least one of its components is differentiable.<ref>{{Cite journal |last=Morayne |first=Michał |date=1987 |title=On differentiability of Peano type functions |url=https://eudml.org/doc/264945 |journal=Colloquium Mathematicum |volume=53 |issue=1 |pages=129–132 |doi=10.4064/cm-53-1-129-132 |issn=0010-1354|doi-access=free }}</ref>