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and space-filling curves to obtain an Osgood curve. For instance, Knopp's construction involves recursively splitting triangles into pairs of smaller triangles, meeting at a shared vertex, by removing triangular wedges. When each level of this construction removes the same fraction of the area of its
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Example of an Osgood curve, constructed by recursively removing wedges from triangles. The wedge angles shrink exponentially, as does the fraction of area removed in each level, leaving nonzero area in the final
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134:, who found a curve that has positive area in every neighborhood of each of its points, based on an earlier construction of
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467:, American Mathematical Society Colloquium Publications, vol. 30, American Mathematical Society, New York, p. 157,
130:). Both examples have positive area in parts of the curve, but zero area in other parts; this flaw was corrected by
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Lance, Timothy; Thomas, Edward (1991), "Arcs with positive measure and a space-filling curve",
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is defined to be an Osgood curve when it is non-self-intersecting (that is, it is either a
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Another way to construct an Osgood curve is to form a two-dimensional version of the
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and totally disconnected subset of the plane is a subset of a Jordan curve.
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It is possible to modify the recursive construction of certain
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282:"On uncountable unions and intersections of measurable sets"
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102:, covering all of the points of the plane, or of any
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114:The first examples of Osgood curves were found by
404:Transactions of the American Mathematical Society
174:point set with non-zero area, and then apply the
106:of the plane, would lead to self-intersections.
23:relating stress to strain in material science.
378:Bulletin de la Société Mathématique de France
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16:Non-self-intersecting curve of positive area
543:, Universitext, New York: Springer-Verlag,
401:(1903), "A Jordan Curve of Positive Area",
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280:Balcerzak, M.; Kharazishvili, A. (1999),
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264:Balcerzak & Kharazishvili (1999)
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585:Knopp's Osgood Curve Construction
418:10.1090/S0002-9947-1903-1500628-5
588:, Wolfram Demonstrations Project
318:Archiv der Mathematik und Physik
59:. Osgood curves are named after
493:The Mathematical Intelligencer
1:
333:American Mathematical Monthly
286:Georgian Mathematical Journal
19:Not to be confused with the
374:"Sur le problème des aires"
155:triangles, the result is a
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43:is a non-self-intersecting
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549:10.1007/978-1-4612-0871-6
249:Lance & Thomas (1991)
178:according to which every
168:Smith–Volterra–Cantor set
67:Definition and properties
116:William Fogg Osgood
104:two-dimensional region
53:two-dimensional region
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37:mathematical analysis
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540:Space-filling curves
537:Sagan, Hans (1994),
298:10.1515/GMJ.1999.201
176:Denjoy–Riesz theorem
172:totally disconnected
96:space-filling curves
57:space-filling curves
21:Ramberg–Osgood curve
92:Hausdorff dimension
90:Osgood curves have
61:William Fogg Osgood
506:10.1007/BF03024322
399:Osgood, William F.
124:Henri Lebesgue
47:that has positive
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391:10.24033/bsmf.694
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41:Osgood curve
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459:RadĂł, Tibor
384:: 197–203,
198:RadĂł (1948)
140:convex hull
604:Categories
592:20 October
530:0795.54022
435:34.0533.02
273:References
94:two, like
81:Jordan arc
522:122497728
427:0002-9947
324:: 103–115
314:Knopp, K.
461:(1948),
372:(1903),
212:, p. 131
152:fractals
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451:1500628
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180:bounded
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110:History
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439:JSTOR
350:JSTOR
186:Notes
79:or a
45:curve
39:, an
615:Area
594:2013
553:ISBN
469:ISBN
423:ISSN
170:, a
128:1903
120:1903
49:area
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