27:
154:
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
30:
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
163:. Instead, reducing the fraction of area removed per level, rapidly enough to leave a constant fraction of the area unremoved, produces an Osgood curve.
399:
134:, who found a curve that has positive area in every neighborhood of each of its points, based on an earlier construction of
463:, American Mathematical Society Colloquium Publications, vol. 30, American Mathematical Society, New York, p. 157,
552:
468:
167:
130:). Both examples have positive area in parts of the curve, but zero area in other parts; this flaw was corrected by
20:
488:
103:
52:
138:. Knopp's example has the additional advantage that its area can be made arbitrarily close to the area of its
328:
280:
Balcerzak, M.; Kharazishvili, A. (1999), "On uncountable unions and intersections of measurable sets",
579:
175:
326:
Lance, Timothy; Thomas, Edward (1991), "Arcs with positive measure and a space-filling curve",
135:
458:
232:
605:
36:
312:(1917), "Einheitliche Erzeugung und Darstellung der Kurven von Peano, Osgood und von Koch",
562:
509:
478:
446:
357:
301:
171:
75:
is defined to be an Osgood curve when it is non-self-intersecting (that is, it is either a
525:
430:
8:
394:
115:
95:
91:
60:
56:
513:
434:
345:
99:
413:
548:
517:
464:
418:
166:
Another way to construct an Osgood curve is to form a two-dimensional version of the
156:
540:
521:
497:
426:
408:
381:
337:
289:
84:
558:
505:
474:
442:
353:
297:
83:) and it has positive area. More formally, it must have positive two-dimensional
72:
365:
160:
123:
544:
293:
610:
599:
422:
454:
182:
and totally disconnected subset of the plane is a subset of a Jordan curve.
309:
76:
486:
Sagan, Hans (1993), "A geometrization of
Lebesgue's space-filling curve",
179:
139:
501:
438:
349:
80:
386:
534:
51:. Despite its area, it is not possible for such a curve to cover any
341:
257:
151:
150:
It is possible to modify the recursive construction of certain
26:
369:
44:
48:
231:, Section 8.3, The Osgood Curves of SierpĂnski and Knopp,
279:
263:
102:, covering all of the points of the plane, or of any
98:. However, they cannot be space-filling curves: by
114:The first examples of Osgood curves were found by
400:Transactions of the American Mathematical Society
174:point set with non-zero area, and then apply the
597:
106:of the plane, would lead to self-intersections.
191:
23:relating stress to strain in material science.
374:Bulletin de la Société Mathématique de France
66:
16:Non-self-intersecting curve of positive area
539:, Universitext, New York: Springer-Verlag,
397:(1903), "A Jordan Curve of Positive Area",
325:
248:
412:
385:
364:
220:
218:
127:
25:
598:
393:
119:
532:
485:
308:
252:
244:
228:
224:
215:
209:
131:
453:
264:Balcerzak & Kharazishvili (1999)
197:
13:
577:
14:
622:
581:Knopp's Osgood Curve Construction
571:
414:10.1090/S0002-9947-1903-1500628-5
584:, Wolfram Demonstrations Project
314:Archiv der Mathematik und Physik
59:. Osgood curves are named after
145:
489:The Mathematical Intelligencer
238:
203:
1:
329:American Mathematical Monthly
282:Georgian Mathematical Journal
272:
19:Not to be confused with the
7:
370:"Sur le problème des aires"
155:triangles, the result is a
55:, distinguishing them from
43:is a non-self-intersecting
10:
627:
109:
18:
545:10.1007/978-1-4612-0871-6
249:Lance & Thomas (1991)
178:according to which every
168:Smith–Volterra–Cantor set
67:Definition and properties
185:
294:10.1023/A:1022102312024
116:William Fogg Osgood
104:two-dimensional region
53:two-dimensional region
32:
37:mathematical analysis
29:
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
33:
387:10.24033/bsmf.694
136:Wacław Sierpiński
618:
592:
591:
589:
578:Dickau, Robert,
565:
528:
481:
449:
416:
390:
389:
360:
321:
304:
267:
261:
255:
242:
236:
233:pp. 136–140
222:
213:
207:
201:
195:
85:Lebesgue measure
626:
625:
621:
620:
619:
617:
616:
615:
596:
595:
587:
585:
574:
569:
555:
471:
460:Length and Area
342:10.2307/2323941
275:
270:
262:
258:
243:
239:
223:
216:
208:
204:
196:
192:
188:
148:
112:
100:Netto's theorem
73:Euclidean plane
71:A curve in the
69:
24:
17:
12:
11:
5:
624:
614:
613:
608:
594:
593:
573:
572:External links
570:
568:
567:
553:
530:
483:
469:
451:
407:(1): 107–112,
391:
362:
336:(2): 124–127,
323:
306:
288:(3): 201–212,
276:
274:
271:
269:
268:
256:
237:
214:
202:
189:
187:
184:
161:Koch snowflake
157:CesĂ ro fractal
147:
144:
111:
108:
68:
65:
15:
9:
6:
4:
3:
2:
623:
612:
609:
607:
604:
603:
601:
583:
582:
576:
575:
564:
560:
556:
554:0-387-94265-3
550:
546:
542:
538:
537:
531:
527:
523:
519:
515:
511:
507:
503:
499:
495:
491:
490:
484:
480:
476:
472:
470:9780821846216
466:
462:
461:
456:
452:
448:
444:
440:
436:
432:
428:
424:
420:
415:
410:
406:
402:
401:
396:
392:
388:
383:
379:
376:(in French),
375:
371:
367:
363:
359:
355:
351:
347:
343:
339:
335:
331:
330:
324:
319:
315:
311:
307:
303:
299:
295:
291:
287:
283:
278:
277:
265:
260:
254:
250:
246:
241:
234:
230:
226:
221:
219:
211:
206:
199:
194:
190:
183:
181:
177:
173:
169:
164:
162:
158:
153:
143:
141:
137:
133:
129:
125:
121:
117:
107:
105:
101:
97:
93:
88:
86:
82:
78:
74:
64:
62:
58:
54:
50:
46:
42:
38:
28:
22:
606:Plane curves
586:, retrieved
580:
535:
496:(4): 37–43,
493:
487:
459:
404:
398:
377:
373:
366:Lebesgue, H.
333:
327:
317:
313:
285:
281:
259:
253:Sagan (1993)
245:Knopp (1917)
240:
229:Sagan (1994)
225:Knopp (1917)
210:Sagan (1994)
205:
193:
165:
159:such as the
149:
146:Construction
132:Knopp (1917)
113:
89:
77:Jordan curve
70:
41:Osgood curve
40:
34:
455:RadĂł, Tibor
380:: 197–203,
198:RadĂł (1948)
140:convex hull
600:Categories
588:20 October
526:0795.54022
431:34.0533.02
273:References
94:two, like
81:Jordan arc
518:122497728
423:0002-9947
320:: 103–115
310:Knopp, K.
457:(1948),
368:(1903),
212:, p. 131
152:fractals
563:1299533
510:1240667
479:0024511
447:1500628
439:1986455
358:1089456
350:2323941
302:1679442
180:bounded
126: (
118: (
110:History
561:
551:
524:
516:
508:
477:
467:
445:
437:
429:
421:
356:
348:
300:
122:) and
31:curve.
514:S2CID
435:JSTOR
346:JSTOR
186:Notes
79:or a
45:curve
39:, an
611:Area
590:2013
549:ISBN
465:ISBN
419:ISSN
170:, a
128:1903
120:1903
49:area
541:doi
522:Zbl
498:doi
427:JFM
409:doi
382:doi
338:doi
290:doi
35:In
602::
559:MR
557:,
547:,
520:,
512:,
506:MR
504:,
494:15
492:,
475:MR
473:,
443:MR
441:,
433:,
425:,
417:,
403:,
378:31
372:,
354:MR
352:,
344:,
334:98
332:,
318:26
316:,
298:MR
296:,
284:,
251:;
247:;
227:;
217:^
142:.
87:.
63:.
566:.
543::
529:.
500::
482:.
450:.
411::
405:4
384::
361:.
340::
322:.
305:.
292::
286:6
266:.
235:.
200:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
