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, doi:10.1090/S0002-9947-1903-1500628-5, ISSN 0002-9947, JFM 34.0533.02, JSTOR 1986455, MR 1500628 .
• Sagan, Hans (1993), "A geometrization of Lebesgue’s space-filling curve", The Mathematical Intelligencer 15 (4): 37–43, doi:10.1007/BF03024322, MR 1240667, Zbl 0795.54022 .
• Dickau, Robert, Knopp's Osgood Curve Construction, Wolfram Demonstrations Project, retrieved 20 October 2013