Peano curve

A Peano curveMathworldPlanetmath or space-filling curve is a continuous mapping of a closed intervalMathworldPlanetmath onto a square.

Such mappings, introduced by Peano in 1890, played an important role in the development of topologyMathworldPlanetmath as a counterexample to the naive ideas of dimensionMathworldPlanetmathPlanetmath — while it is true that one cannot map a space onto a space of higher dimension using a smooth mapMathworldPlanetmath, this is no longer true if one only requires continuity as opposed to smoothness. The Peano curve and similar examples led to a rethinking of the foundations of topology and analysisMathworldPlanetmath, and underscored the importance of formulating topological notions in a rigorous fashion.

However, still, a space-filling curve cannot ever be one-to-one; otherwise invariance of dimension would be violated.

Many space-filling curves may be obtained as the limit of a sequence, hnn, of continuous functionsMathworldPlanetmath hn:[0,1][0,1]×[0,1]. Figure 1 (\PMlinktofilesource, showing the ranges of the first few approximations to Hilbert’s space-filling curve, illustrates a common case in which each successive approximation is obtained by applying a recursive procedure to its predecessor.

Figure 1: The ranges of the first six approximations to Hilbert’s space-filling curve
Title Peano curve
Canonical name PeanoCurve
Date of creation 2013-03-22 16:32:29
Last modified on 2013-03-22 16:32:29
Owner stevecheng (10074)
Last modified by stevecheng (10074)
Numerical id 13
Author stevecheng (10074)
Entry type Definition
Classification msc 28A80
Synonym space-filling curve
Synonym space filling curve