Editing Peano curve (section)

{{Short description|Space-filling curve}}
{{about|a particular curve defined by Giuseppe Peano|other curves with similar properties|space-filling curve}}
[[Image:Peanocurve.svg|400px|thumb|Three iterations of a Peano curve construction, whose limit is a space-filling curve.]]
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 | image1 = Peano 1.GIF
 | image2 = Peano 2.GIF
 | footer = Two iterations of a Peano curve
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In [[geometry]], the '''Peano curve''' is the first example of a [[space-filling curve]] to be discovered, by [[Giuseppe Peano]] in 1890.<ref>{{citation|first=G.|last=Peano|authorlink=Giuseppe Peano|title=Sur une courbe, qui remplit toute une aire plane|journal=[[Mathematische Annalen]]|volume=36|issue=1|year=1890|pages=157–160|doi=10.1007/BF01199438}}.</ref> Peano's curve is a [[Surjective function|surjective]], [[continuous function]] from the [[unit interval]] [[onto]] the [[unit square]], however it is not [[Injective function|injective]]. Peano was motivated by an earlier result of [[Georg Cantor]] that these two sets have the same [[cardinality]]. Because of this example, some authors use the phrase "Peano curve" to refer more generally to any space-filling curve.<ref>{{citation|title=Differential Geometry|first=Heinrich Walter|last=Gugenheimer|publisher=Courier Dover Publications|year=1963|isbn=9780486157207|page=3|url=https://books.google.com/books?id=CSYtkV4NTioC&pg=PA}}.</ref>