# Peano curve

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

Such mappings, introduced by Peano in 1890, played an important role in the development of topology as a counterexample to the naive ideas of dimension — while it is true that one cannot map a space onto a space of higher dimension using a smooth map, 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 analysis, 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, $\langle\,h_{n}\mid n\in\mathbb{N}\,\rangle$, of continuous functions $h_{n}\colon[0,1]\to[0,1]\times[0,1]$. Figure 1 (\PMlinktofilesource codehilbert.cc), 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.

Title Peano curve PeanoCurve 2013-03-22 16:32:29 2013-03-22 16:32:29 stevecheng (10074) stevecheng (10074) 13 stevecheng (10074) Definition msc 28A80 space-filling curve space filling curve