135:
815:
794:
848:
1394:
864:
871:
783:
1165:
1148:
827:
808:
38:
836:
1551:
1382:
878:
1054:, the 2-fold axes are through the midpoints of opposite edges, and the number of them is half the number of edges. The other axes are through opposite vertices and through centers of opposite faces, except in the case of the tetrahedron, where the 3-fold axes are each through one vertex and the center of one face.
1338:
3-fold rotational symmetry at one point and 2-fold at another one (or ditto in 3D with respect to parallel axes) implies rotation group p6, i.e. double translational symmetry and 6-fold rotational symmetry at some point (or, in 3D, parallel axis). The translation distance for the symmetry generated
1480:
1389:, an example of p6, , (632) (with colors) and p6m, , (*632) (without colors); the lines are reflection axes if colors are ignored, and a special kind of symmetry axis if colors are not ignored: reflection reverts the colors. Rectangular line grids in three orientations can be distinguished.
1250:
2-fold rotocenters (including possible 4-fold and 6-fold), if present at all, form the translate of a lattice equal to the translational lattice, scaled by a factor 1/2. In the case translational symmetry in one dimension, a similar property applies, though the term "lattice" does not
1122:
In 4D, continuous or discrete rotational symmetry about a plane corresponds to corresponding 2D rotational symmetry in every perpendicular plane, about the point of intersection. An object can also have rotational symmetry about two perpendicular planes, e.g. if it is the
1334:
Scaling of a lattice divides the number of points per unit area by the square of the scale factor. Therefore, the number of 2-, 3-, 4-, and 6-fold rotocenters per primitive cell is 4, 3, 2, and 1, respectively, again including 4-fold as a special case of 2-fold, etc.
162:, is the property a shape has when it looks the same after some rotation by a partial turn. An object's degree of rotational symmetry is the number of distinct orientations in which it looks exactly the same for each rotation.
1169:
Arrangement within a primitive cell of 2-, 3-, and 6-fold rotocenters, alone or in combination (consider the 6-fold symbol as a combination of a 2- and a 3-fold symbol); in the case of 2-fold symmetry only, the shape of the
146:
has rotational symmetry because it appears the same when rotated by one third of a full turn about its center. Because its appearance is identical in three distinct orientations, its rotational symmetry is
1254:
3-fold rotocenters (including possible 6-fold), if present at all, form a regular hexagonal lattice equal to the translational lattice, rotated by 30° (or equivalently 90°), and scaled by a factor
542:
1326:
1288:
413:
1364:
489:
1229:
the rotation group of any lattice (every lattice is upside-down the same, but that does not apply for this symmetry); it is e.g. the rotation group of the
1481:
Topological Bound States in the
Continuum in Arrays of Dielectric Spheres. By Dmitrii N. Maksimov, LV Kirensky Institute of Physics, Krasnoyarsk, Russia
1292:
4-fold rotocenters, if present at all, form a regular square lattice equal to the translational lattice, rotated by 45°, and scaled by a factor
1565:
773:
758:
743:
728:
713:
1119:). An example of approximate spherical symmetry is the Earth (with respect to density and other physical and chemical properties).
169:
rotated 90°, however the only geometric objects that are fully rotationally symmetric at any angle are spheres, circles and other
102:
946:
should be distinguished: there are other symmetry groups of the same abstract group type which are geometrically different, see
488:
should be distinguished: there are other symmetry groups of the same abstract group type which are geometrically different, see
74:
17:
1330:
6-fold rotocenters, if present at all, form a regular hexagonal lattice which is the translate of the translational lattice.
427:°, etc.) does not change the object. A "1-fold" symmetry is no symmetry (all objects look alike after a rotation of 360°).
1555:
1424:
81:
55:
431:
1230:
270:
For symmetry with respect to rotations about a point we can take that point as origin. These rotations form the special
1533:
1502:
121:
88:
1434:
689:
A typical 3D object with rotational symmetry (possibly also with perpendicular axes) but no mirror symmetry is a
682:
If there is e.g. rotational symmetry with respect to an angle of 100°, then also with respect to one of 20°, the
508:
1295:
1257:
382:
204:
70:
59:
947:
1084:
1580:
1470:
Rotational symmetry of
Weingarten spheres in homogeneous three-manifolds. By Jos ́e A. G ́alvez, Pablo Mira
1088:
1083:(no change when rotating about one axis, or for any rotation). That is, no dependence on the angle using
134:
1590:
819:
329:; in other words, the intersection of the full symmetry group and the group of direct isometries. For
1386:
1189:
212:
253:
1342:
814:
683:
330:
143:
95:
48:
787:
377:, with respect to a particular point (in 2D) or axis (in 3D) means that rotation by an angle of
1449:
1398:
1181:
241:
with respect to all translations, so space is homogeneous, and the symmetry group is the whole
238:
1561:
182:
793:
1067:
895:
with multiple symmetry axes through the same point, there are the following possibilities:
341:
165:
Certain geometric objects are partially symmetrical when rotated at certain angles such as
8:
633:
549:
291:
1493:
1157:
1079:
496:
1585:
1529:
1506:
1498:
1393:
1237:
1124:
1063:
966:
892:
1429:
1199:
Together with double translational symmetry the rotation groups are the following
932:
847:
345:
271:
1219:
1200:
993:
863:
852:
592:
500:
326:
231:
200:
1174:
can be different. For the case p6, a fundamental domain is indicated in yellow.
574:(as divided along the flag's diagonal and rotated about the flag's center point)
1419:
1193:
1153:
1051:
908:
452:
216:
870:
782:
1574:
1208:
1171:
1164:
1147:
856:
629:
1521:
1185:
1012:
801:
567:
460:
1444:
1128:
1016:
958:
1510:
1469:
1439:
1092:
989:
962:
798:
588:
571:
139:
1244:
1212:
1104:
1062:
Rotational symmetry with respect to any angle is, in two dimensions,
936:
690:
340:
if they do not distinguish different directions in space. Because of
170:
1127:
of two rotationally symmetry 2D figures, as in the case of e.g. the
826:
807:
344:, the rotational symmetry of a physical system is equivalent to the
37:
1414:
1132:
1112:
650:
613:
337:
208:
192:
188:
159:
27:
Property of objects which appear unchanged after a partial rotation
1138:
1216:
1111:(i.e. rotational symmetry with respect to a central axis) like a
1007:
6×5-fold, 10×3-fold, and 15×2-fold axes: the rotation group
939:. Although the same notation is used, the geometric and abstract
835:
676:
668:
459:. For each point or axis of symmetry, the abstract group type is
455:
is specified by the point or axis of symmetry, together with the
166:
1057:
886:
237:
Symmetry with respect to all rotations about all points implies
1550:
980:
3×4-fold, 4×3-fold, and 6×2-fold axes: the rotation group
566:; letters Z, N, S; the outlines, albeit not the colors, of the
1401:, an example of (732) symmetry and , (*732) (without colors)
1116:
841:
303:
1381:
1243:
p6 (632): 1×6-fold, 2×3-fold, 3×2-fold; rotation group of a
877:
985:
1107:
which refer to an object having cylindrical symmetry, or
1095:
through the axis, and a radial half-line, respectively.
1300:
1262:
1233:
with the equilateral triangles alternatingly colored.
513:
387:
310:
In another definition of the word, the rotation group
1345:
1298:
1260:
511:
385:
1019:. The group is isomorphic to alternating group
1046:(rotational symmetries like prisms and antiprisms).
333:objects it is the same as the full symmetry group.
62:. Unsourced material may be challenged and removed.
1358:
1320:
1282:
1236:p4 (442): 2×4-fold, 2×2-fold; rotation group of a
536:
407:
1572:
1180:2-fold rotational symmetry together with single
1139:Rotational symmetry with translational symmetry
953:4×3-fold and 3×2-fold axes: the rotation group
351:
1192:of a rotation. There are two rotocenters per
1058:Rotational symmetry with respect to any angle
887:Multiple symmetry axes through the same point
254:modified notion of symmetry for vector fields
474:. Although for the latter also the notation
931:). This is the rotation group of a regular
537:{\displaystyle {\tfrac {360^{\circ }}{n}}.}
1321:{\displaystyle {\tfrac {1}{2}}{\sqrt {2}}}
1283:{\displaystyle {\tfrac {1}{3}}{\sqrt {3}}}
1142:
408:{\displaystyle {\tfrac {360^{\circ }}{n}}}
1528:. Princeton: Princeton University Press.
1207:p2 (2222): 4×2-fold; rotation group of a
122:Learn how and when to remove this message
1087:and no dependence on either angle using
219:of rotational symmetry is a subgroup of
133:
1073:In three dimensions we can distinguish
14:
1573:
1497:, Wiley-Interscience, New York, p.2.
187:Formally the rotational symmetry is
1425:Crystallographic restriction theorem
1028:. The group contains 10 versions of
481:is used, the geometric and abstract
417:(180°, 120°, 90°, 72°, 60°, 51
371:discrete rotational symmetry of the
60:adding citations to reliable sources
31:
1339:by one such pair of rotocenters is
663:is the rotation group of a regular
176:
24:
653:, computer-generated (CG), ceiling
357:Rotational symmetry of order
25:
1602:
1543:
1549:
1435:Point groups in three dimensions
1392:
1380:
1203:, with axes per primitive cell:
1163:
1156:of 2- and 4-fold rotocenters. A
1146:
876:
869:
862:
846:
834:
825:
813:
806:
792:
781:
36:
907:perpendicular 2-fold axes: the
256:the symmetry group can also be
47:needs additional citations for
1520:
1485:
1474:
1463:
1091:. The fundamental domain is a
1066:. The fundamental domain is a
957:of order 12 of a regular
948:dihedral symmetry groups in 3D
13:
1:
1456:
992:. The group is isomorphic to
314:is the symmetry group within
1562:Rotational Symmetry Examples
1368:
1359:{\displaystyle 2{\sqrt {3}}}
700:
548:Examples without additional
490:cyclic symmetry groups in 3D
352:Discrete rotational symmetry
191:with respect to some or all
7:
1407:
696:
301:this is the rotation group
10:
1607:
1378:
1190:fixed, or invariant, point
832:
779:
327:group of direct isometries
180:
1387:Hexakis triangular tiling
1231:regular triangular tiling
840:The starting position in
632:(this one has additional
367:-fold rotational symmetry
1160:is indicated in yellow.
294:with determinant 1. For
1085:cylindrical coordinates
788:Double Pendulum fractal
684:greatest common divisor
671:in 2D and of a regular
1450:Translational symmetry
1399:Order 3-7 kisrhombille
1366:times their distance.
1360:
1322:
1284:
1188:. A rotocenter is the
1182:translational symmetry
1011:of order 60 of a
984:of order 24 of a
538:
409:
239:translational symmetry
148:
18:Rotationally symmetric
1361:
1323:
1285:
1152:Arrangement within a
1089:spherical coordinates
595:; sometimes the term
539:
410:
183:Rotational invariance
137:
71:"Rotational symmetry"
1558:at Wikimedia Commons
1343:
1296:
1258:
1225:p3 (333): 3×3-fold;
1131:and various regular
1075:cylindrical symmetry
509:
383:
56:improve this article
1581:Rotational symmetry
1556:Rotational symmetry
1491:Loeb, A.L. (1971).
1050:In the case of the
634:reflection symmetry
597:trilateral symmetry
550:reflection symmetry
338:are SO(3)-invariant
292:orthogonal matrices
152:Rotational symmetry
1494:Color and Symmetry
1356:
1318:
1309:
1280:
1271:
1158:fundamental domain
1080:spherical symmetry
1037:and 6 versions of
899:In addition to an
686:of 100° and 360°.
534:
529:
497:fundamental domain
438:-fold symmetry is
405:
403:
348:conservation law.
149:
1591:Binocular rivalry
1554:Media related to
1405:
1404:
1375:Hyperbolic plane
1354:
1316:
1308:
1278:
1270:
1178:
1177:
1125:Cartesian product
1064:circular symmetry
967:alternating group
893:discrete symmetry
884:
883:
528:
402:
342:Noether's theorem
205:direct isometries
142:appearing on the
132:
131:
124:
106:
16:(Redirected from
1598:
1553:
1539:
1513:
1489:
1483:
1478:
1472:
1467:
1430:Lorentz symmetry
1396:
1384:
1372:Euclidean plane
1369:
1365:
1363:
1362:
1357:
1355:
1350:
1327:
1325:
1324:
1319:
1317:
1312:
1310:
1301:
1289:
1287:
1286:
1281:
1279:
1274:
1272:
1263:
1209:parallelogrammic
1201:wallpaper groups
1167:
1150:
1143:
1045:
1036:
1027:
1010:
1003:
976:
956:
945:
930:
923:
916:
906:
902:
880:
873:
866:
850:
838:
829:
817:
810:
796:
785:
771:
756:
741:
726:
711:
701:
674:
666:
662:
644:
623:
607:
582:
561:
545:
543:
541:
540:
535:
530:
524:
523:
514:
487:
480:
473:
466:
458:
450:
446:
437:
426:
425:
421:
416:
414:
412:
411:
406:
404:
398:
397:
388:
374:
366:
360:
346:angular momentum
336:Laws of physics
324:
306:
300:
290:
280:
272:orthogonal group
266:
251:
229:
203:. Rotations are
198:
177:Formal treatment
154:, also known as
144:Isle of Man flag
127:
120:
116:
113:
107:
105:
64:
40:
32:
21:
1606:
1605:
1601:
1600:
1599:
1597:
1596:
1595:
1571:
1570:
1546:
1536:
1517:
1516:
1490:
1486:
1479:
1475:
1468:
1464:
1459:
1454:
1410:
1397:
1385:
1349:
1344:
1341:
1340:
1311:
1299:
1297:
1294:
1293:
1273:
1261:
1259:
1256:
1255:
1168:
1151:
1141:
1060:
1052:Platonic solids
1044:
1038:
1035:
1029:
1026:
1020:
1008:
1002:
996:
994:symmetric group
975:
969:
961:. The group is
954:
944:
940:
925:
918:
915:
911:
909:dihedral groups
904:
900:
889:
855:'s interlocked
853:Snoldelev Stone
851:
839:
820:US Bicentennial
818:
797:
786:
770:
764:
755:
749:
740:
734:
725:
719:
710:
704:
699:
672:
664:
661:
657:
639:
618:
602:
593:Borromean rings
577:
556:
519:
515:
512:
510:
507:
506:
504:
486:
482:
479:
475:
472:
468:
464:
456:
448:
444:
439:
435:
423:
419:
418:
393:
389:
386:
384:
381:
380:
378:
372:
364:
358:
354:
315:
302:
295:
282:
281:, the group of
274:
257:
242:
232:Euclidean group
220:
215:. Therefore, a
201:Euclidean space
196:
185:
179:
156:radial symmetry
128:
117:
111:
108:
65:
63:
53:
41:
28:
23:
22:
15:
12:
11:
5:
1604:
1594:
1593:
1588:
1583:
1569:
1568:
1559:
1545:
1544:External links
1542:
1541:
1540:
1534:
1515:
1514:
1484:
1473:
1461:
1460:
1458:
1455:
1453:
1452:
1447:
1442:
1437:
1432:
1427:
1422:
1420:Axial symmetry
1417:
1411:
1409:
1406:
1403:
1402:
1390:
1377:
1376:
1373:
1353:
1348:
1332:
1331:
1328:
1315:
1307:
1304:
1290:
1277:
1269:
1266:
1252:
1248:
1241:
1234:
1223:
1194:primitive cell
1184:is one of the
1176:
1175:
1161:
1154:primitive cell
1140:
1137:
1101:axisymmetrical
1059:
1056:
1048:
1047:
1042:
1033:
1024:
1005:
1000:
988:and a regular
978:
973:
951:
942:
917:of order
913:
888:
885:
882:
881:
874:
867:
860:
857:drinking horns
844:
831:
830:
823:
811:
804:
790:
778:
777:
768:
762:
753:
747:
738:
732:
723:
717:
708:
698:
695:
659:
655:
654:
637:
616:
600:
575:
533:
527:
522:
518:
484:
477:
470:
463:of order
453:symmetry group
442:
401:
396:
392:
362:, also called
353:
350:
217:symmetry group
178:
175:
130:
129:
44:
42:
35:
26:
9:
6:
4:
3:
2:
1603:
1592:
1589:
1587:
1584:
1582:
1579:
1578:
1576:
1567:
1563:
1560:
1557:
1552:
1548:
1547:
1537:
1535:0-691-02374-3
1531:
1527:
1523:
1522:Weyl, Hermann
1519:
1518:
1512:
1508:
1504:
1503:9780471543350
1500:
1496:
1495:
1488:
1482:
1477:
1471:
1466:
1462:
1451:
1448:
1446:
1443:
1441:
1438:
1436:
1433:
1431:
1428:
1426:
1423:
1421:
1418:
1416:
1413:
1412:
1400:
1395:
1391:
1388:
1383:
1379:
1374:
1371:
1370:
1367:
1351:
1346:
1336:
1329:
1313:
1305:
1302:
1291:
1275:
1267:
1264:
1253:
1249:
1246:
1242:
1239:
1235:
1232:
1228:
1224:
1221:
1218:
1214:
1210:
1206:
1205:
1204:
1202:
1197:
1195:
1191:
1187:
1186:Frieze groups
1183:
1173:
1172:parallelogram
1166:
1162:
1159:
1155:
1149:
1145:
1144:
1136:
1134:
1130:
1126:
1120:
1118:
1114:
1110:
1106:
1102:
1098:
1094:
1090:
1086:
1082:
1081:
1076:
1071:
1069:
1065:
1055:
1053:
1041:
1032:
1023:
1018:
1014:
1006:
999:
995:
991:
987:
983:
979:
972:
968:
964:
960:
952:
949:
938:
935:, or regular
934:
928:
922:
910:
898:
897:
896:
894:
879:
875:
872:
868:
865:
861:
858:
854:
849:
845:
843:
837:
833:
828:
824:
821:
816:
812:
809:
805:
803:
800:
795:
791:
789:
784:
780:
775:
767:
763:
760:
752:
748:
745:
737:
733:
730:
722:
718:
715:
707:
703:
702:
694:
692:
687:
685:
680:
678:
670:
652:
648:
642:
638:
635:
631:
630:Star of David
627:
621:
617:
615:
611:
605:
601:
598:
594:
590:
586:
580:
576:
573:
569:
565:
559:
555:
554:
553:
551:
546:
531:
525:
520:
516:
502:
498:
493:
491:
462:
454:
451:. The actual
445:
433:
428:
399:
394:
390:
376:
368:
361:
349:
347:
343:
339:
334:
332:
328:
322:
318:
313:
308:
305:
298:
293:
289:
285:
278:
273:
268:
264:
260:
255:
249:
245:
240:
235:
233:
227:
223:
218:
214:
210:
206:
202:
199:-dimensional
194:
190:
184:
174:
172:
168:
163:
161:
157:
153:
145:
141:
136:
126:
123:
115:
104:
101:
97:
94:
90:
87:
83:
80:
76:
73: –
72:
68:
67:Find sources:
61:
57:
51:
50:
45:This article
43:
39:
34:
33:
30:
19:
1525:
1492:
1487:
1476:
1465:
1337:
1333:
1226:
1198:
1179:
1121:
1108:
1100:
1097:Axisymmetric
1096:
1078:
1074:
1072:
1061:
1049:
1039:
1030:
1021:
1013:dodecahedron
997:
981:
970:
926:
920:
903:-fold axis,
890:
802:traffic sign
765:
750:
735:
720:
705:
688:
681:
656:
649:, Octagonal
646:
640:
625:
619:
609:
603:
596:
584:
578:
570:symbol; the
568:yin and yang
563:
562:, 180°: the
557:
547:
494:
461:cyclic group
440:
429:
370:
363:
356:
355:
335:
320:
316:
312:of an object
311:
309:
296:
287:
283:
276:
269:
262:
258:
247:
243:
236:
225:
221:
186:
164:
155:
151:
150:
118:
109:
99:
92:
85:
78:
66:
54:Please help
49:verification
46:
29:
1566:Math Is Fun
1445:Space group
1213:rectangular
1129:duocylinder
1109:axisymmetry
1017:icosahedron
959:tetrahedron
252:. With the
213:orientation
211:preserving
147:three-fold.
1575:Categories
1457:References
1440:Screw axis
1105:adjectives
1093:half-plane
990:octahedron
963:isomorphic
799:Roundabout
589:triskelion
572:Union Flag
447:or simply
209:isometries
181:See also:
140:triskelion
82:newspapers
1524:(1982) .
1245:hexagonal
1133:duoprisms
1068:half-line
937:bipyramid
691:propeller
521:∘
395:∘
193:rotations
171:spheroids
112:June 2018
1586:Symmetry
1526:Symmetry
1415:Ambigram
1408:See also
1247:lattice.
1240:lattice.
1113:doughnut
697:Examples
651:muqarnas
614:swastika
599:is used;
583:, 120°:
432:notation
375:th order
207:, i.e.,
189:symmetry
160:geometry
1220:lattice
1217:rhombic
1015:and an
859:design
679:in 3D.
677:pyramid
675:-sided
669:polygon
667:-sided
645:, 45°:
624:, 60°:
608:, 90°:
544:
505:
422:⁄
415:
379:
167:squares
96:scholar
1532:
1511:163904
1509:
1501:
1251:apply.
1238:square
1215:, and
610:tetrad
501:sector
331:chiral
325:, the
98:
91:
84:
77:
69:
1564:from
1117:torus
933:prism
842:shogi
822:Star
647:octad
626:hexad
585:triad
499:is a
369:, or
304:SO(3)
230:(see
103:JSTOR
89:books
1530:ISBN
1507:OCLC
1499:ISBN
1103:are
1099:and
1077:and
986:cube
891:For
774:more
759:more
744:more
729:more
714:more
564:dyad
495:The
434:for
430:The
138:The
75:news
1227:not
965:to
929:≥ 2
643:= 8
622:= 6
606:= 4
581:= 3
560:= 2
517:360
503:of
391:360
299:= 3
275:SO(
234:).
195:in
158:in
58:by
1577::
1505:,
1211:,
1196:.
1135:.
1070:.
776:)
761:)
746:)
731:)
716:)
693:.
628:,
612:,
591:,
587:,
552::
492:.
467:,
307:.
286:×
267:.
173:.
1538:.
1352:3
1347:2
1314:2
1306:2
1303:1
1276:3
1268:3
1265:1
1222:.
1115:(
1043:5
1040:D
1034:3
1031:D
1025:5
1022:A
1009:I
1004:.
1001:4
998:S
982:O
977:.
974:4
971:A
955:T
950:.
943:n
941:D
927:n
924:(
921:n
919:2
914:n
912:D
905:n
901:n
772:(
769:6
766:C
757:(
754:5
751:C
742:(
739:4
736:C
727:(
724:3
721:C
712:(
709:2
706:C
673:n
665:n
660:n
658:C
641:n
636:)
620:n
604:n
579:n
558:n
532:.
526:n
485:n
483:C
478:n
476:C
471:n
469:Z
465:n
457:n
449:n
443:n
441:C
436:n
424:7
420:3
400:n
373:n
365:n
359:n
323:)
321:n
319:(
317:E
297:m
288:m
284:m
279:)
277:m
265:)
263:m
261:(
259:E
250:)
248:m
246:(
244:E
228:)
226:m
224:(
222:E
197:m
125:)
119:(
114:)
110:(
100:·
93:·
86:·
79:·
52:.
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
