Osgood curve

Osgood curve: In mathematics, an Osgood curve is a Jordan curve of positive area. The first example was found by Osgood (1903).

Examples of Osgood curves can be produced by slightly modifying one of the constructions of space-filling curves with image the unit square to make it an embedding, though the cost is that it no longer fills the whole unit square.

References

Osgood, William F. (1903), "A Jordan Curve of Positive Area", Transactions of the American Mathematical Society 4: 107–112, ISSN 0002-9947, JSTOR 1986455 .

Sagan, Hans (1993), "A geometrization of Lebesgue’s space-filling curve", The Mathematical Intelligencer 15 (4): 37–43, doi:10.1007/BF03024322 .

http://demonstrations.wolfram.com/KnoppsOsgoodCurveConstruction/

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Osgood curve

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