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, $\\langle\\,h_{n}\\mid n\\in\\mathbb{N}\\,\\rangle$ , 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.

单词	PeanoCurve
释义	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, $\\langle\\,h_{n}\\mid n\\in\\mathbb{N}\\,\\rangle$ , 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. Figure 1: The ranges of the first six approximations to Hilbert’s space-filling curve
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