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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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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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Balcerzak, M.; Kharazishvili, A. (1999), "On uncountable unions and intersections of measurable sets",
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Lance, Timothy; Thomas, Edward (1991), "Arcs with positive measure and a space-filling curve",
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312:(1917), "Einheitliche Erzeugung und Darstellung der Kurven von Peano, Osgood und von Koch",
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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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Sagan, Hans (1993), "A geometrization of
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It is possible to modify the recursive construction of certain
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231:, Section 8.3, The Osgood Curves of SierpĂnski and Knopp,
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400:Transactions of the American Mathematical Society
174:point set with non-zero area, and then apply the
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106:of the plane, would lead to self-intersections.
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23:relating stress to strain in material science.
374:Bulletin de la Société Mathématique de France
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16:Non-self-intersecting curve of positive area
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397:(1903), "A Jordan Curve of Positive Area",
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581:Knopp's Osgood Curve Construction
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414:10.1090/S0002-9947-1903-1500628-5
584:, Wolfram Demonstrations Project
314:Archiv der Mathematik und Physik
59:. Osgood curves are named after
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489:The Mathematical Intelligencer
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329:American Mathematical Monthly
282:Georgian Mathematical Journal
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19:Not to be confused with the
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370:"Sur le problème des aires"
155:triangles, the result is a
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249:Lance & Thomas (1991)
178:according to which every
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67:Definition and properties
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294:10.1023/A:1022102312024
116:William Fogg Osgood
104:two-dimensional region
53:two-dimensional region
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37:mathematical analysis
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536:Space-filling curves
533:Sagan, Hans (1994),
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
502:10.1007/BF03024322
395:Osgood, William F.
124:Henri Lebesgue
47:that has positive
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146:Construction
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41:Osgood curve
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455:RadĂł, Tibor
380:: 197–203,
198:RadĂł (1948)
140:convex hull
600:Categories
588:20 October
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273:References
94:two, like
81:Jordan arc
518:122497728
423:0002-9947
320:: 103–115
310:Knopp, K.
457:(1948),
368:(1903),
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152:fractals
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180:bounded
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435:JSTOR
346:JSTOR
186:Notes
79:or a
45:curve
39:, an
611:Area
590:2013
549:ISBN
465:ISBN
419:ISSN
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128:1903
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49:area
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