Editing Space-filling curve (section)

==History==
In 1890, [[Giuseppe Peano]] discovered a continuous curve, now called the [[Peano curve]], that passes through every point of the unit square.{{sfn|Peano|1890}} 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 (see {{section link|#Properties}} below).

It was common to associate the vague notions of ''thinness'' and 1-dimensionality to curves; all normally encountered curves were [[piecewise]] differentiable (that is, have piecewise continuous derivatives), 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]] (for any positive integer ''n''). It was also easy to extend Peano's example to continuous curves without endpoints, which filled the entire ''n''-dimensional Euclidean space (where ''n'' is 2, 3, or any other positive integer).

Most well-known space-filling curves are constructed iteratively as the limit of a sequence of [[piecewise linear function|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 expansion]]s 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 [[proof without words|graphical visualization]] was motivated by a desire for a completely rigorous proof owing nothing to pictures.  At that time (the beginning of the foundation of general topology), 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.{{sfn|Hilbert|1891}} 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.

[[File:Hilbert curve.svg|400px|thumb|Six iterations of the Hilbert curve construction, whose limiting space-filling curve was devised by mathematician [[David Hilbert]].]]