339:
89:
53:
968:
237:
191:
145:
27:
979:
424:
946:
957:
862:
832:
892:
802:
885:
855:
825:
795:
327:
notation is used. The terms horizontal (h) and vertical (v) imply the existence and direction of reflections with respect to a vertical axis of symmetry. Also shown are
630:-fold improper rotation axis, i.e., the symmetry group contains a combination of a reflection in the horizontal plane and a rotation by an angle 180°/n. Thus, like
1092:
1073:
1057:
1044:
997:
93:
732:. Rotations become translations in the limit. Portions of the infinite plane can also be cut and connected into an infinite cylinder.
1027:
31:
275:
265:
255:
229:
219:
209:
183:
173:
163:
127:
117:
107:
81:
71:
45:
289:
18:
1080:
1064:
270:
260:
224:
214:
178:
168:
122:
112:
76:
1068:, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995,
324:
972:
338:
1076:
766:
665:
241:
149:
511:
500:
195:
983:
580:
506:
1019:
1112:
1088:
1069:
1053:
1040:
1023:
756:
623:
332:
637:, it contains a number of improper rotations without containing the corresponding rotations.
1011:
761:
694:
328:
135:
88:
52:
967:
608:
312:
236:
190:
144:
26:
961:
950:
308:
704:
applies e.g. for a rectangular tile with its top side different from its bottom side.
1106:
1012:
409:
884:
854:
824:
751:
736:
713:
469:
392:
320:
978:
794:
423:
575:(1*). It has vertical mirror planes. This is the symmetry group for a regular
428:
300:-fold rotational or reflectional symmetry about one axis (by an angle of 360°/
956:
945:
861:
831:
285:
891:
801:
422:
337:
693:
of order 4 are two of the three 3D symmetry group types with the
1052:
2008, John H. Conway, Heidi
Burgiel, Chaim Goodman-Strauss,
1010:
Sands, Donald E. (1993). "Crystal
Systems and Geometry".
712:
In the limit these four groups represent
Euclidean plane
1018:. Mineola, New York: Dover Publications, Inc. p.
611:, for which the same notation is used; abstract group
342:
Example symmetry subgroup tree for dihedral symmetry:
904:
15:
1039:, 2003, John Horton Conway and Derek A. Smith
514:with respect to a plane perpendicular to the
8:
734:
998:Dihedral symmetry in three dimensions
7:
1063:Kaleidoscopes: Selected Writings of
288:, there are four infinite series of
304:) that does not change the object.
331:in brackets, and, in parentheses,
14:
977:
966:
955:
944:
890:
883:
860:
853:
830:
823:
800:
793:
290:point groups in three dimensions
273:
268:
263:
258:
253:
235:
227:
222:
217:
212:
207:
189:
181:
176:
171:
166:
161:
143:
125:
120:
115:
110:
105:
87:
79:
74:
69:
51:
43:
25:
19:point groups in three dimensions
1099:, 11.5 Spherical Coxeter groups
1014:Introduction to Crystallography
1085:Geometries and Transformations
1:
1037:On Quaternions and Octonions
387:-fold rotational symmetry -
319:= ∞ they correspond to four
1129:
923:
745:
742:
607:(not to be confused with
134:
1050:The Symmetries of Things
973:Elongated square pyramid
544:full acro-n-gonal group
1097:Finite symmetry groups
488:=1 this is denoted by
439:
350:
426:
341:
284:In three dimensional
32:Involutional symmetry
518:-fold rotation axis.
427:Piece of loose-fill
307:They are the finite
242:Icosahedral symmetry
150:Tetrahedral symmetry
739:
697:as abstract group.
655:), also denoted by
626:axis, also called 2
512:reflection symmetry
501:reflection symmetry
466:ortho-n-gonal group
196:Octahedral symmetry
21:
984:Pentagonal pyramid
774:Cylindrical (n=6)
735:
666:inversion symmetry
605:gyro-n-gonal group
540:pyramidal symmetry
507:bilateral symmetry
462:prismatic symmetry
440:
389:acro-n-gonal group
351:
16:
1093:978-1-107-10340-5
1074:978-0-471-01003-6
1058:978-1-56881-220-5
1045:978-1-56881-134-5
989:
988:
898:
897:
568:=1 we have again
562:biradial symmetry
333:orbifold notation
282:
281:
94:Dihedral symmetry
1120:
1033:
1017:
981:
970:
959:
948:
905:
894:
887:
864:
857:
834:
827:
804:
797:
771:Euclidean plane
740:
695:Klein four-group
609:symmetric groups
546:(abstract group
468:(abstract group
391:(abstract group
329:Coxeter notation
278:
277:
276:
272:
271:
267:
266:
262:
261:
257:
256:
239:
232:
231:
230:
226:
225:
221:
220:
216:
215:
211:
210:
193:
186:
185:
184:
180:
179:
175:
174:
170:
169:
165:
164:
147:
136:Polyhedral group
130:
129:
128:
124:
123:
119:
118:
114:
113:
109:
108:
91:
84:
83:
82:
78:
77:
73:
72:
55:
48:
47:
46:
29:
22:
1128:
1127:
1123:
1122:
1121:
1119:
1118:
1117:
1103:
1102:
1030:
1009:
1006:
994:
982:
971:
960:
949:
937:
928:
919:
912:
903:
880:
850:
820:
790:
731:
727:
723:
719:
710:
702:
689:
679:
661:
649:
635:
616:
591:
573:
558:
553:); in biology
551:
526:
494:
482:
474:
448:
436:
397:
371:
356:
347:
309:symmetry groups
274:
269:
264:
259:
254:
252:
250:
248:
244:
240:
228:
223:
218:
213:
208:
206:
204:
202:
198:
194:
182:
177:
172:
167:
162:
160:
158:
156:
152:
148:
126:
121:
116:
111:
106:
104:
102:
100:
96:
92:
80:
75:
70:
68:
66:
64:
60:
58:Cyclic symmetry
56:
44:
42:
40:
38:
34:
30:
12:
11:
5:
1126:
1124:
1116:
1115:
1105:
1104:
1101:
1100:
1078:
1065:H.S.M. Coxeter
1060:
1047:
1034:
1028:
1005:
1002:
1001:
1000:
993:
990:
987:
986:
975:
964:
962:Square pyramid
953:
951:Parallelepiped
941:
940:
935:
931:
926:
922:
917:
910:
902:
899:
896:
895:
888:
881:
878:
875:
873:
870:
866:
865:
858:
851:
848:
845:
843:
840:
836:
835:
828:
821:
818:
815:
813:
810:
806:
805:
798:
791:
788:
785:
783:
780:
776:
775:
772:
769:
764:
759:
754:
748:
747:
744:
729:
725:
721:
717:
709:
706:
700:
687:
677:
672:
671:
670:
669:
659:
647:
633:
624:rotoreflection
614:
589:
584:
571:
556:
549:
524:
519:
492:
480:
472:
446:
434:
421:
420:
418:
414:
413:
395:
369:
363:
362:
360:
355:
352:
345:
280:
279:
246:
233:
200:
187:
154:
140:
139:
132:
131:
98:
85:
62:
49:
36:
13:
10:
9:
6:
4:
3:
2:
1125:
1114:
1111:
1110:
1108:
1098:
1094:
1090:
1086:
1082:
1079:
1077:
1075:
1071:
1067:
1066:
1061:
1059:
1055:
1051:
1048:
1046:
1042:
1038:
1035:
1031:
1029:0-486-67839-3
1025:
1021:
1016:
1015:
1008:
1007:
1003:
999:
996:
995:
991:
985:
980:
976:
974:
969:
965:
963:
958:
954:
952:
947:
943:
942:
938:
932:
929:
920:
913:
907:
906:
900:
893:
889:
886:
882:
876:
874:
871:
868:
867:
863:
859:
856:
852:
846:
844:
841:
838:
837:
833:
829:
826:
822:
816:
814:
811:
808:
807:
803:
799:
796:
792:
786:
784:
781:
778:
777:
773:
770:
768:
765:
763:
760:
758:
755:
753:
750:
749:
741:
738:
737:Frieze groups
733:
715:
714:frieze groups
708:Frieze groups
707:
705:
703:
696:
692:
690:
682:
680:
667:
663:
662:
654:
650:
643:
639:
638:
636:
629:
625:
621:
618:); It has a 2
617:
610:
606:
602:
598:
596:
592:
585:
582:
578:
574:
567:
563:
559:
552:
545:
541:
537:
533:
531:
527:
520:
517:
513:
509:
508:
503:
502:
497:
495:
487:
483:
476:
475:
467:
463:
459:
455:
453:
449:
442:
441:
437:
430:
425:
419:
416:
415:
411:
410:trivial group
407:
403:
399:
398:
390:
386:
382:
378:
376:
372:
365:
364:
361:
358:
357:
353:
348:
340:
336:
334:
330:
326:
322:
321:frieze groups
318:
314:
310:
305:
303:
299:
295:
291:
287:
243:
238:
234:
197:
192:
188:
151:
146:
142:
141:
137:
133:
95:
90:
86:
59:
54:
50:
33:
28:
24:
23:
20:
1096:
1095:Chapter 11:
1084:
1081:N.W. Johnson
1062:
1049:
1036:
1013:
933:
924:
915:
908:
711:
698:
685:
684:
675:
674:
673:
657:
656:
652:
645:
641:
631:
627:
619:
612:
604:
600:
594:
587:
586:
576:
569:
565:
561:
554:
547:
543:
539:
535:
529:
522:
521:
515:
505:
499:
490:
489:
485:
478:
470:
465:
461:
457:
451:
444:
443:
432:
405:
401:
393:
388:
384:
380:
374:
367:
366:
343:
316:
306:
301:
297:
293:
283:
57:
644:=1 we have
498:and called
406:no symmetry
138:, , (*n32)
1004:References
767:Schönflies
743:Notations
664:; this is
599:of order 2
560:is called
534:of order 2
456:of order 2
429:cushioning
349:, , (*224)
325:Schönflies
1087:, (2018)
746:Examples
691:, , (*22)
510:. It has
379:of order
296:≥1) with
17:Selected
1113:Symmetry
1107:Category
992:See also
901:Examples
757:Orbifold
438:symmetry
286:geometry
249:, (*532)
203:, (*432)
157:, (*332)
101:, (*n22)
939:(*55):
930:(*44):
762:Coxeter
728:, and S
681:, (2*)
581:pyramid
579:-sided
504:, also
484:); for
417:Achiral
400:); for
65:, (*nn)
1091:
1072:
1056:
1043:
1026:
921:(1x):
622:-fold
564:. For
528:, , (*
359:Chiral
315:. For
593:, , (
450:, , (
431:with
373:, , (
354:Types
311:on a
39:, (*)
1089:ISBN
1070:ISBN
1054:ISBN
1041:ISBN
1024:ISBN
869:p11g
839:p11m
809:p1m1
716:as C
683:and
640:for
496:(1*)
404:=1:
313:cone
1020:165
812:*∞∞
752:IUC
724:, C
720:, C
548:Dih
542:or
479:Dih
464:or
1109::
1083::
1022:.
936:5v
927:4v
872:∞×
849:∞h
842:∞*
819:∞v
782:∞∞
779:p1
726:∞v
722:∞h
701:2v
688:2v
678:2h
653:1×
634:nd
615:2n
603:-
597:×)
590:2n
557:2v
538:-
530:nn
525:nv
477:×
460:-
454:*)
447:nh
435:2h
383:-
375:nn
346:4h
335:.
323:.
251:=
205:=
159:=
103:=
99:nh
67:=
63:nv
41:=
1032:.
934:C
925:C
918:i
916:C
914:/
911:2
909:S
879:∞
877:S
847:C
817:C
789:∞
787:C
730:∞
718:∞
699:C
686:C
676:C
668:.
660:i
658:C
651:(
648:2
646:S
642:n
632:D
628:n
620:n
613:Z
601:n
595:n
588:S
583:.
577:n
572:s
570:C
566:n
555:C
550:n
536:n
532:)
523:C
516:n
493:s
491:C
486:n
481:1
473:n
471:Z
458:n
452:n
445:C
433:C
412:)
408:(
402:n
396:n
394:Z
385:n
381:n
377:)
370:n
368:C
344:D
317:n
302:n
298:n
294:n
292:(
247:h
245:I
201:h
199:O
155:d
153:T
97:D
61:C
37:s
35:C
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