Osgood curve

In mathematics, an Osgood curve is a non-self-intersecting curve (either a Jordan curve or a Jordan arc) of positive area.[1] More formally, these are curves in the Euclidean plane with positive two-dimensional Lebesgue measure.

Not to be confused with the Ramberg–Osgood curve relating stress to strain in material science.
Fractal construction of an Osgood curve by recursively removing wedges from triangles. As the wedges narrow, the fraction of area removed decreases exponentially, so the area remaining in the final curve is nonzero.

History

The first examples of Osgood curves were found by William Fogg Osgood (1903) and Henri Lebesgue (1903). Both examples have positive area in parts of the curve, but zero area in other parts; this flaw was corrected by Knopp (1917), who found a curve that has positive area in every neighborhood of each of its points, based on an earlier construction of Wacław Sierpiński. Knopp's example has the additional advantage that its area can be controlled to be any desired fraction of the area of its convex hull.[2]

Fractal construction

Although most space-filling curves are not Osgood curves (they have positive area but often include infinitely many self-intersections, failing to be Jordan curves) it is possible to modify the recursive construction of space-filling curves or other fractal curves to obtain an Osgood curve.[3] For instance, Knopp's construction involves recursively splitting triangles into pairs of smaller triangles, meeting at a shared vertex, by removing triangular wedges. When the removed wedges at each level of this construction cover the same fraction of the area of their triangles, the result is a Cesàro fractal such as the Koch snowflake, but removing wedges whose areas shrink more rapidly produces an Osgood curve.[2]

Denjoy–Riesz construction

Another way to construct an Osgood curve is to form a two-dimensional version of the Smith–Volterra–Cantor set, a totally disconnected point set with non-zero area, and then apply the Denjoy–Riesz theorem according to which every bounded and totally disconnected subset of the plane is a subset of a Jordan curve.[4]

Notes
  1. Radó (1948).
  2. Knopp (1917); Sagan (1994), Section 8.3, The Osgood Curves of Sierpínski and Knopp, pp. 136–140.
  3. Knopp (1917); Lance & Thomas (1991); Sagan (1993)).
  4. Balcerzak & Kharazishvili (1999).
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References
  • Balcerzak, M.; Kharazishvili, A. (1999), "On uncountable unions and intersections of measurable sets", Georgian Mathematical Journal, 6 (3): 201–212, doi:10.1023/A:1022102312024, MR 1679442.
  • Knopp, K. (1917), "Einheitliche Erzeugung und Darstellung der Kurven von Peano, Osgood und von Koch", Archiv der Mathematik und Physik, 26: 103–115.
  • Lance, Timothy; Thomas, Edward (1991), "Arcs with positive measure and a space-filling curve", American Mathematical Monthly, 98 (2): 124–127, doi:10.2307/2323941, JSTOR 2323941, MR 1089456.
  • Lebesgue, H. (1903), "Sur le problème des aires", Bulletin de la Société Mathématique de France (in French), 31: 197–203, doi:10.24033/bsmf.694
  • Osgood, William F. (1903), "A Jordan Curve of Positive Area", Transactions of the American Mathematical Society, 4 (1): 107–112, doi:10.1090/S0002-9947-1903-1500628-5, ISSN 0002-9947, JFM 34.0533.02, JSTOR 1986455, MR 1500628.
  • Radó, Tibor (1948), Length and Area, American Mathematical Society Colloquium Publications, vol. 30, American Mathematical Society, New York, p. 157, ISBN 9780821846216, MR 0024511.
  • 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.
  • Sagan, Hans (1994), Space-filling curves, Universitext, New York: Springer-Verlag, doi:10.1007/978-1-4612-0871-6, ISBN 0-387-94265-3, MR 1299533.
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