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 animportant role in the development of topology as a counterexampleto the naive ideas of dimension — while it istrue that one cannot map a space onto a space of higher dimension using asmooth map, this is no longer true if one only requires continuity as opposed tosmoothness. The Peano curve and similar examples led to a rethinking of the foundationsof topology and analysis, and underscored the importance of formulatingtopological 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, $⟨h_n\mid n\in \mathbb{N}⟩$, of continuous functions $h_n:[0,1]\to [0,1]\times [0,1]$. Figure 1 (source 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.

Figure 1: The ranges of the first six approximations to Hilbert’s space-filling curve