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Deleted incorrect statements about the Osgood curve. Although it has positive area, it does not fill space and it does not contain a square. Tag: Reverted |
Reverted 1 edit by David Radcliffe (talk): True that Osborne's curve is not a square filling curve, but that does not mean that there are not square filling curves |
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{{anchor|Jordan}}A plane simple closed curve is also called a '''Jordan curve'''. It is also defined as a non-self-intersecting [[loop (topology)|continuous loop]] in the plane.<ref>{{Cite book|url=https://books.google.com/books?id=0Q9mbXCQRyoC&pg=PA7|title=Depth, Crossings and Conflicts in Discrete Geometry|last=Sulovský|first=Marek|date=2012|publisher=Logos Verlag Berlin GmbH| isbn=9783832531195|page=7|language=en}}</ref> The [[Jordan curve theorem]] states that the [[set complement]] in a plane of a Jordan curve consists of two [[connected component (topology)|connected component]]s (that is the curve divides the plane in two non-intersecting [[region (mathematics)|regions]] that are both connected).
The definition of a curve includes figures that can hardly be called curves in common usage. For example, the image of a simple curve can cover a [[Square (geometry)|square]] in the plane ([[space-filling curve]]) and thus have a positive area.<ref>{{cite journal|last=Osgood|first=William F.|date=January 1903|title=A Jordan Curve of Positive Area|journal=Transactions of the American Mathematical Society|publisher=[[American Mathematical Society]]|volume=4|issue=1|pages=107–112|doi=10.2307/1986455|issn=0002-9947|jstor=1986455|author-link1=William Fogg Osgood|doi-access=free}}<!--|access-date=2008-06-04--></ref> [[Fractal curve]]s can have properties that are strange for the common sense. For example, a fractal curve can have a [[Hausdorff dimension]] bigger than one (see [[Koch snowflake]]) and even a positive area. An example is the [[dragon curve]], which has many other unusual properties.
==Differentiable curve==
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