83:
125:
31:
235:(which is the 3-manifold obtained by gluing two copies of the horned sphere together along the corresponding points of their boundaries) is in fact the 3-sphere. One can consider other gluings of the solid horned sphere to a copy of itself, arising from different homeomorphisms of the boundary sphere to itself. This has also been shown to be the 3-sphere. The solid Alexander horned sphere is an example of a
136:
removed results. This embedding extends to a continuous map from the whole sphere, which is injective (hence a topological embedding since the sphere is compact) since points in the sphere approaching two different points of the Cantor set will end up in different 'horns' at some stage and therefore
1156:
247:
One can generalize
Alexander's construction to generate other horned spheres by increasing the number of horns at each stage of Alexander's construction or considering the analogous construction in higher dimensions.
161:
simply connected, unlike the exterior of the usual round sphere; a loop linking a torus in the above construction cannot be shrunk to a point without touching the horned sphere. This shows that the
383:
1174:
58:). It is a particular topological embedding of a two-dimensional sphere in three-dimensional space. Together with its inside, it is a topological
471:
620:
251:
Other substantially different constructions exist for constructing such "wild" spheres. Another example, also found by
Alexander, is
653:
535:
513:
162:
82:
573:
124:
86:
Diagram of the first few iterative steps in the construction of
Alexander's horned sphere, from Alexander's original 1924 paper
1180:
1036:
993:
1239:
1114:
1244:
1229:
1196:
680:
378:
252:
205:
51:
43:
846:
190:
170:
132:
By considering only the points of the tori that are not removed at some stage, an embedding of the sphere with a
702:
165:
does not hold in three dimensions, as
Alexander had originally thought. Alexander also proved that the theorem
1139:
186:
174:
1188:
747:
613:
178:
113:
Connect a standard punctured torus to each side of the cut, interlinked with the torus on the other side.
973:
665:
324:
256:
1134:
1129:
919:
851:
392:
182:
91:
381:(1924), "An Example of a Simply Connected Surface Bounding a Region which is not Simply Connected",
322:
Bing, R. H. (1952), "A homeomorphism between the 3-sphere and the sum of two solid horned spheres",
892:
869:
804:
752:
737:
670:
272:
1119:
1099:
1063:
1058:
821:
451:
418:
349:
216:
146:
59:
588:
1234:
1162:
1124:
1048:
956:
861:
767:
742:
732:
675:
658:
648:
643:
606:
553:
531:
509:
467:
436:
410:
341:
232:
17:
1079:
946:
929:
757:
556:
459:
426:
400:
333:
177:
embeddings. This is one of the earliest examples where the need for distinction between the
154:
67:
481:
361:
70:; i.e., every loop can be shrunk to a point while staying inside. However, the exterior is
1094:
1031:
692:
577:
477:
357:
209:
99:
789:
396:
239:; i.e., a closed complementary domain of the embedding of a 2-sphere into the 3-sphere.
30:
1109:
1053:
1041:
1012:
968:
951:
934:
887:
831:
816:
784:
722:
523:
431:
103:
1223:
963:
939:
809:
779:
762:
727:
712:
494:
487:
236:
571:
1208:
1203:
1104:
1084:
841:
774:
447:
157:; i.e., every loop can be shrunk to a point while staying inside. The exterior is
1169:
1089:
799:
794:
502:
384:
Proceedings of the
National Academy of Sciences of the United States of America
1022:
1007:
1002:
983:
717:
583:
260:
228:
133:
414:
345:
978:
924:
836:
687:
561:
277:
197:
440:
879:
282:
224:
201:
47:
593:
405:
909:
826:
629:
353:
463:
897:
422:
95:
337:
123:
81:
29:
74:
simply connected, unlike the exterior of the usual round sphere.
528:
A comprehensive introduction to differential geometry (Volume 1)
602:
145:
The horned sphere, together with its inside, is a topological
90:
The
Alexander horned sphere is the particular (topological)
598:
570:
Zbigniew
Fiedorowicz. Math 655 – Introduction to Topology.
456:
Mathematical
Omnibus. 30 Lectures on Classical Mathematics
102:
obtained by the following construction, starting with a
500:
Hocking, John
Gilbert; Young, Gail Sellers (1988) .
1148:
1072:
1021:
992:
908:
878:
860:
701:
636:
501:
495:http://pi.math.cornell.edu/~hatcher/AT/ATpage.html
458:, Providence, RI: American Mathematical Society,
219:of the non-simply connected domain is called the
594:PC OpenGL demo rendering and expanding the cusp
614:
196:Now consider Alexander's horned sphere as an
8:
305:
223:. Although the solid horned sphere is not a
128:Animated construction of Alexander's sphere.
116:Repeat steps 1–2 on the two tori just added
621:
607:
599:
430:
404:
391:(1), National Academy of Sciences: 8–10,
55:
1175:List of fractals by Hausdorff dimension
298:
309:
7:
584:Construction of the Alexander sphere
110:Remove a radial slice of the torus.
259:, a pathological embedding of the
25:
1157:How Long Is the Coast of Britain?
1181:The Fractal Geometry of Nature
18:Alexander's horned sphere
1:
221:solid Alexander horned sphere
169:hold in three dimensions for
1197:Chaos: Making a New Science
557:"Alexander's Horned Sphere"
1261:
206:one-point compactification
191:piecewise linear manifolds
163:Jordan–Schönflies theorem
306:Hocking & Young 1988
187:differentiable manifolds
253:Antoine's horned sphere
137:have different images.
40:Alexander horned sphere
34:Alexander horned sphere
1189:The Beauty of Fractals
129:
87:
35:
530:. Publish or Perish.
325:Annals of Mathematics
208:of the 3-dimensional
183:topological manifolds
151:Alexander horned ball
127:
85:
64:Alexander horned ball
33:
27:Topological embedding
1135:Lewis Fry Richardson
1130:Hamid Naderi Yeganeh
920:Burning Ship fractal
852:Weierstrass function
308:, pp. 175–176.
255:, which is based on
204:, considered as the
1240:Eponyms in geometry
893:Space-filling curve
870:Multifractal system
753:Space-filling curve
738:Sierpinski triangle
492:Algebraic Topology,
406:10.1073/pnas.10.1.8
397:1924PNAS...10....8A
285:, specifically the
273:Cantor tree surface
263:into the 3-sphere.
52:J. W. Alexander
1245:1924 introductions
1230:Geometric topology
1120:Aleksandr Lyapunov
1100:Desmond Paul Henry
1064:Self-avoiding walk
1059:Percolation theory
703:Iterated function
644:Fractal dimensions
589:rotating animation
576:2005-08-25 at the
554:Weisstein, Eric W.
452:Tabachnikov, Serge
257:Antoine's necklace
130:
88:
36:
1217:
1216:
1163:Coastline paradox
1140:Wacław Sierpiński
1125:Benoit Mandelbrot
1049:Fractal landscape
957:Misiurewicz point
862:Strange attractor
743:Apollonian gasket
733:Sierpinski carpet
473:978-0-8218-4316-1
328:, Second Series,
193:became apparent.
98:in 3-dimensional
16:(Redirected from
1252:
1080:Michael Barnsley
947:Lyapunov fractal
805:Sierpiński curve
758:Blancmange curve
623:
616:
609:
600:
567:
566:
541:
519:
507:
484:
443:
434:
408:
379:Alexander, J. W.
365:
364:
319:
313:
303:
231:showed that its
171:piecewise linear
155:simply connected
141:Impact on theory
68:simply connected
21:
1260:
1259:
1255:
1254:
1253:
1251:
1250:
1249:
1220:
1219:
1218:
1213:
1144:
1095:Felix Hausdorff
1068:
1032:Brownian motion
1017:
988:
911:
904:
874:
856:
847:Pythagoras tree
704:
697:
693:Self-similarity
637:Characteristics
632:
627:
580:– Lecture notes
578:Wayback Machine
552:
551:
548:
538:
524:Spivak, Michael
522:
516:
499:
474:
464:10.1090/mbk/046
446:
377:
374:
369:
368:
338:10.2307/1969804
321:
320:
316:
304:
300:
295:
269:
245:
243:Generalizations
210:Euclidean space
143:
100:Euclidean space
80:
28:
23:
22:
15:
12:
11:
5:
1258:
1256:
1248:
1247:
1242:
1237:
1232:
1222:
1221:
1215:
1214:
1212:
1211:
1206:
1201:
1193:
1185:
1177:
1172:
1167:
1166:
1165:
1152:
1150:
1146:
1145:
1143:
1142:
1137:
1132:
1127:
1122:
1117:
1112:
1110:Helge von Koch
1107:
1102:
1097:
1092:
1087:
1082:
1076:
1074:
1070:
1069:
1067:
1066:
1061:
1056:
1051:
1046:
1045:
1044:
1042:Brownian motor
1039:
1028:
1026:
1019:
1018:
1016:
1015:
1013:Pickover stalk
1010:
1005:
999:
997:
990:
989:
987:
986:
981:
976:
971:
969:Newton fractal
966:
961:
960:
959:
952:Mandelbrot set
949:
944:
943:
942:
937:
935:Newton fractal
932:
922:
916:
914:
906:
905:
903:
902:
901:
900:
890:
888:Fractal canopy
884:
882:
876:
875:
873:
872:
866:
864:
858:
857:
855:
854:
849:
844:
839:
834:
832:Vicsek fractal
829:
824:
819:
814:
813:
812:
807:
802:
797:
792:
787:
782:
777:
772:
771:
770:
760:
750:
748:Fibonacci word
745:
740:
735:
730:
725:
723:Koch snowflake
720:
715:
709:
707:
699:
698:
696:
695:
690:
685:
684:
683:
678:
673:
668:
663:
662:
661:
651:
640:
638:
634:
633:
628:
626:
625:
618:
611:
603:
597:
596:
591:
586:
581:
568:
547:
546:External links
544:
543:
542:
536:
520:
514:
497:
488:Hatcher, Allen
485:
472:
444:
373:
370:
367:
366:
332:(2): 354–362,
314:
297:
296:
294:
291:
290:
289:
280:
275:
268:
265:
244:
241:
142:
139:
122:
121:
114:
111:
104:standard torus
79:
76:
50:discovered by
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
1257:
1246:
1243:
1241:
1238:
1236:
1233:
1231:
1228:
1227:
1225:
1210:
1207:
1205:
1202:
1199:
1198:
1194:
1191:
1190:
1186:
1183:
1182:
1178:
1176:
1173:
1171:
1168:
1164:
1161:
1160:
1158:
1154:
1153:
1151:
1147:
1141:
1138:
1136:
1133:
1131:
1128:
1126:
1123:
1121:
1118:
1116:
1113:
1111:
1108:
1106:
1103:
1101:
1098:
1096:
1093:
1091:
1088:
1086:
1083:
1081:
1078:
1077:
1075:
1071:
1065:
1062:
1060:
1057:
1055:
1052:
1050:
1047:
1043:
1040:
1038:
1037:Brownian tree
1035:
1034:
1033:
1030:
1029:
1027:
1024:
1020:
1014:
1011:
1009:
1006:
1004:
1001:
1000:
998:
995:
991:
985:
982:
980:
977:
975:
972:
970:
967:
965:
964:Multibrot set
962:
958:
955:
954:
953:
950:
948:
945:
941:
940:Douady rabbit
938:
936:
933:
931:
928:
927:
926:
923:
921:
918:
917:
915:
913:
907:
899:
896:
895:
894:
891:
889:
886:
885:
883:
881:
877:
871:
868:
867:
865:
863:
859:
853:
850:
848:
845:
843:
840:
838:
835:
833:
830:
828:
825:
823:
820:
818:
815:
811:
810:Z-order curve
808:
806:
803:
801:
798:
796:
793:
791:
788:
786:
783:
781:
780:Hilbert curve
778:
776:
773:
769:
766:
765:
764:
763:De Rham curve
761:
759:
756:
755:
754:
751:
749:
746:
744:
741:
739:
736:
734:
731:
729:
728:Menger sponge
726:
724:
721:
719:
716:
714:
713:Barnsley fern
711:
710:
708:
706:
700:
694:
691:
689:
686:
682:
679:
677:
674:
672:
669:
667:
664:
660:
657:
656:
655:
652:
650:
647:
646:
645:
642:
641:
639:
635:
631:
624:
619:
617:
612:
610:
605:
604:
601:
595:
592:
590:
587:
585:
582:
579:
575:
572:
569:
564:
563:
558:
555:
550:
549:
545:
539:
537:0-914098-70-5
533:
529:
525:
521:
517:
515:0-486-65676-4
511:
506:
505:
498:
496:
493:
489:
486:
483:
479:
475:
469:
465:
461:
457:
453:
449:
448:Fuchs, Dmitry
445:
442:
438:
433:
428:
424:
420:
416:
412:
407:
402:
398:
394:
390:
386:
385:
380:
376:
375:
371:
363:
359:
355:
351:
347:
343:
339:
335:
331:
327:
326:
318:
315:
311:
307:
302:
299:
292:
288:
287:Fox–Artin arc
284:
281:
279:
276:
274:
271:
270:
266:
264:
262:
258:
254:
249:
242:
240:
238:
237:crumpled cube
234:
230:
226:
222:
218:
214:
211:
207:
203:
199:
194:
192:
188:
184:
180:
176:
172:
168:
164:
160:
156:
152:
148:
140:
138:
135:
126:
119:
115:
112:
109:
108:
107:
105:
101:
97:
93:
84:
77:
75:
73:
69:
65:
61:
57:
53:
49:
45:
41:
32:
19:
1209:Chaos theory
1204:Kaleidoscope
1195:
1187:
1179:
1105:Gaston Julia
1085:Georg Cantor
910:Escape-time
842:Gosper curve
790:LĂ©vy C curve
775:Dragon curve
654:Box-counting
560:
527:
503:
491:
455:
388:
382:
329:
323:
317:
312:, p. 55
301:
286:
250:
246:
220:
212:
195:
166:
158:
153:, and so is
150:
144:
131:
118:ad infinitum
117:
89:
78:Construction
71:
66:, and so is
63:
44:pathological
39:
37:
1200:(1987 book)
1192:(1986 book)
1184:(1982 book)
1170:Fractal art
1090:Bill Gosper
1054:LĂ©vy flight
800:Peano curve
795:Moore curve
681:Topological
666:Correlation
310:Spivak 1999
1224:Categories
1008:Orbit trap
1003:Buddhabrot
996:techniques
984:Mandelbulb
785:Koch curve
718:Cantor set
293:References
261:Cantor set
229:R. H. Bing
179:categories
134:Cantor set
46:object in
1115:Paul LĂ©vy
994:Rendering
979:Mandelbox
925:Julia set
837:Hexaflake
768:Minkowski
688:Recursion
671:Hausdorff
562:MathWorld
508:. Dover.
415:0027-8424
372:Citations
346:0003-486X
278:Wild knot
200:into the
198:embedding
92:embedding
1235:Fractals
1025:fractals
912:fractals
880:L-system
822:T-square
630:Fractals
574:Archived
526:(1999).
504:Topology
454:(2007),
441:16576780
283:Wild arc
267:See also
225:manifold
202:3-sphere
48:topology
974:Tricorn
827:n-flake
676:Packing
659:Higuchi
649:Assouad
482:2350979
432:1085500
393:Bibcode
362:0049549
354:1969804
217:closure
54: (
1073:People
1023:Random
930:Filled
898:H tree
817:String
705:system
534:
512:
480:
470:
439:
429:
421:
413:
360:
352:
344:
233:double
215:. The
189:, and
175:smooth
149:, the
147:3-ball
96:sphere
62:, the
60:3-ball
1149:Other
423:84202
419:JSTOR
350:JSTOR
94:of a
42:is a
532:ISBN
510:ISBN
468:ISBN
437:PMID
411:ISSN
342:ISSN
167:does
56:1924
38:The
490:,
460:doi
427:PMC
401:doi
334:doi
181:of
159:not
72:not
1226::
1159:"
559:.
478:MR
476:,
466:,
450:;
435:,
425:,
417:,
409:,
399:,
389:10
387:,
358:MR
356:,
348:,
340:,
330:56
227:,
185:,
106::
1155:"
622:e
615:t
608:v
565:.
540:.
518:.
462::
403::
395::
336::
213:R
173:/
120:.
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
