An Osgood curve is a simple curve in the plane having positive two-dimensional Lebesgue measure.
Osgood (1903) constructed the first such Jordan curve,
demonstrating that a simple curve in the plane
need not have area zero.
Every Osgood curve has Hausdorff dimension 2, but unlike a plane-filling function,
it contains no open set in the plane.
Thus positive area does not imply that a curve
fills a two-dimensional region. Knopp (1917) later gave
a unified recursive construction related to the Peano curve and Koch snowflake.
See also
Area,
Hausdorff Dimension,
Jordan Curve,
Lebesgue
Measure,
Plane-Filling Function,
Simple Curve
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References
Knopp, K. "Einheitliche Erzeugung und Darstellung der Kurven von Peano, Osgood und v. Koch." Arch. f. Math. u. Phys. 26,
103-115, 1917.Osgood, W. F. "A Jordan Curve of Positive Area."
Trans. Amer. Math. Soc. 4, 107-112, 1903. https://doi.org/10.1090/S0002-9947-1903-1500628-5.
Pegg, E. Jr. Mathematical Games. Episode 16: "Space-Filling Curves."
Apr. 18, 2024. https://www.youtube.com/watch?v=3qbZadltTCI.
Companion notebook: https://community.wolfram.com/t/23888.Sagan,
H. "Jordan Curves of Positive Lebesgue Measure." Ch. 8 in Space-Filling
Curves. New York: Springer-Verlag, pp. 131-143, 1994.
Cite this as:
Weisstein, Eric W. "Osgood Curve." From
MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/OsgoodCurve.html
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