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