Space-filling curve
In mathematical analysis, a space-filling curve is a curve whose range contains the entire 2-dimensional unit square. Because Giuseppe Peano was the first to discover one, space-filling curves in the 2-dimensional plane are sometimes called Peano curves, but that phrase also refers to the Peano curve, the specific example of a space-filling curve found by Peano.
Definition
Intuitively, a continuous curve in two or three dimensions can be thought of as the path of a continuously moving point. To eliminate the inherent vagueness of this notion, Jordan in 1887 introduced the following rigorous definition, which has since been adopted as the precise description of the notion of a continuous curve:In the most general form, the range of such a function may lie in an arbitrary topological space, but in the most commonly studied cases, the range will lie in a Euclidean space such as the 2-dimensional plane or the 3-dimensional space.
Sometimes, the curve is identified with the image of the function, instead of the function itself. It is also possible to define curves without endpoints to be a continuous function on the real line.
History
In 1890, Peano discovered a continuous curve, now called the Peano curve, that passes through every point of the unit square. His purpose was to construct a continuous mapping from the unit interval onto the unit square. Peano was motivated by Georg Cantor's earlier counterintuitive result that the infinite number of points in a unit interval is the same cardinality as the infinite number of points in any finite-dimensional manifold, such as the unit square. The problem Peano solved was whether such a mapping could be continuous; i.e., a curve that fills a space. Peano's solution does not set up a continuous one-to-one correspondence between the unit interval and the unit square, and indeed such a correspondence does not exist.It was common to associate the vague notions of thinness and 1-dimensionality to curves; all normally encountered curves were piecewise differentiable, and such curves cannot fill up the entire unit square. Therefore, Peano's space-filling curve was found to be highly counterintuitive.
From Peano's example, it was easy to deduce continuous curves whose ranges contained the n-dimensional hypercube. It was also easy to extend Peano's example to continuous curves without endpoints, which filled the entire n-dimensional Euclidean space.
Most well-known space-filling curves are constructed iteratively as the limit of a sequence of piecewise linear continuous curves, each one more closely approximating the space-filling limit.
Peano's ground-breaking article contained no illustrations of his construction, which is defined in terms of ternary expansions and a mirroring operator. But the graphical construction was perfectly clear to him—he made an ornamental tiling showing a picture of the curve in his home in Turin. Peano's article also ends by observing that the technique can be obviously extended to other odd bases besides base 3. His choice to avoid any appeal to graphical visualization was, no doubt, motivated by a desire for a well-founded, completely rigorous proof owing nothing to pictures. At that time, graphical arguments were still included in proofs, yet were becoming a hindrance to understanding often counterintuitive results.
A year later, David Hilbert published in the same journal a variation of Peano's construction. Hilbert's article was the first to include a picture helping to visualize the construction technique, essentially the same as illustrated here. The analytic form of the Hilbert curve, however, is more complicated than Peano's.
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Outline of the construction of a space-filling curve
Let denote the Cantor space.We start with a continuous function from the Cantor space onto the entire unit interval. From it, we get a continuous function from the topological product onto the entire unit square by setting
Since the Cantor set is homeomorphic to the product, there is a continuous bijection from the Cantor set onto. The composition of and is a continuous function mapping the Cantor set onto the entire unit square.
Finally, one can extend to a continuous function whose domain is the entire unit interval. This can be done either by using the Tietze extension theorem on each of the components of, or by simply extending "linearly".
Properties
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 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 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. 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 curves, but they are not space-filling.
For the classic Peano and Hilbert space-filling curves, where two subcurves intersect, there is self-contact without self-crossing. A space-filling curve can be 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 curves. No differentiable space-filling curve can exist. Roughly speaking, differentiability puts a bound on how fast the curve can turn.
The Hahn–Mazurkiewicz theorem
The Hahn–Mazurkiewicz theorem is the following characterization of spaces that are the continuous image of curves:Spaces that are the continuous image of a unit interval are sometimes called Peano spaces.
In many formulations of the Hahn–Mazurkiewicz theorem, second-countable is replaced by metrizable. These two formulations are equivalent. In one direction a compact Hausdorff space is a normal space and, by the Urysohn metrization theorem, second-countable then implies metrizable. Conversely, a compact metric space is second-countable.
Kleinian groups
There are many natural examples of space-filling, or rather sphere-filling, curves in the theory of doubly degenerate Kleinian groups. For example,showed that the circle at infinity of the universal cover of a fiber of a mapping torus of a pseudo-Anosov map is a sphere-filling curve.