Space-filling Curve - Properties

Properties

A space-filling curve must be everywhere self-intersecting in the technical sense that the curve is not injective. 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 of the two images is 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.

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 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 constructions. No differentiable space-filling curve can exist. Roughly speaking, differentiability puts a bound on how fast the curve can turn.