1177:
31:
320:
1159:
2 (more symmetry of the pattern on the tile does not change that, because of the arrangement of the tiles). The rectangle is a more convenient unit to consider as fundamental domain (or set of two of them) than a parallelogram consisting of part of a tile and part of another one.
1142:
Alternatively, e.g. a rectangle may define the whole object, even if the translation vectors are not perpendicular, if it has two sides parallel to one translation vector, while the other translation vector starting at one side of the rectangle ends at the opposite side.
1163:
In 2D there may be translational symmetry in one direction for vectors of any length. One line, not in the same direction, fully defines the whole object. Similarly, in 3D there may be translational symmetry in one or two directions for vectors of any length. One plane
1146:
For example, consider a tiling with equal rectangular tiles with an asymmetric pattern on them, all oriented the same, in rows, with for each row a shift of a fraction, not one half, of a tile, always the same, then we have only translational symmetry,
965:, in 2D an infinite strip, and in 3D a slab, such that the vector starting at one side ends at the other side. Note that the strip and slab need not be perpendicular to the vector, hence can be narrower or thinner than the length of the vector.
274:
75:
292:
Translational symmetry of an object means that a particular translation does not change the object. For a given object, the translations for which this applies form a group, the
976:. In particular, the multiplicity may be equal to the dimension. This implies that the object is infinite in all directions. In this case, the set of all translations forms a
208:
125:
737:
785:
1131:
can be represented by complex numbers. For two given lattice points, equivalence of choices of a third point to generate a lattice shape is represented by the
790:
780:
775:
595:
1247:
859:
742:
1111:
themselves are integer linear combinations of the other two vectors. If not, not all translations are possible with the other pair. Each pair
890:
219:
1123:. One parallelogram fully defines the whole object. Without further symmetry, this parallelogram is a fundamental domain. The vectors
752:
281:
are translationally invariant under a spatial translation if they do not distinguish different points in space. According to
747:
727:
1296:
692:
600:
37:
1176:
1306:
732:
1165:
883:
367:
286:
907:, the set of points with the same properties due to the translational symmetry form the infinite discrete set
903:
Translational invariance implies that, at least in one direction, the object is infinite: for any given point
148:
1155:(the same applies without shift). With rotational symmetry of order two of the pattern on the tile we have
687:
650:
618:
605:
171:
152:
1232:
1214:
968:
In spaces with dimension higher than 1, there may be multiple translational symmetries. For each set of
719:
387:
347:
337:
282:
164:
1252:
1199:
1016:
of the symmetry: any pattern on or in it is possible, and this defines the whole object. See also
876:
864:
705:
535:
1013:
636:
626:
186:
80:
296:
of the object, or, if the object has more kinds of symmetry, a subgroup of the symmetry group.
1301:
1237:
1192:
815:
700:
663:
553:
1257:
1227:
1210:
1206:
835:
515:
507:
499:
491:
483:
462:
452:
442:
432:
416:
397:
357:
128:
1242:
1148:
1136:
1017:
977:
820:
573:
558:
329:
278:
216:
doesn't change if the argument function is translated. More precisely it must hold that
1267:
1185:
1005:
993:
985:
981:
840:
825:
658:
563:
293:
1290:
1132:
1120:
548:
377:
17:
30:
1262:
1195:
with subsequent computation of absolute values is a translation-invariant operator.
984:
one is transformed into the other by a matrix of integer coefficients of which the
962:
845:
830:
631:
613:
543:
997:
989:
671:
587:
311:
140:
1000:
of the matrix formed by a set of translation vectors is the hypervolume of the
951:
810:
676:
568:
307:
1180:
The less-than-relation on the real numbers is invariant under translation.
285:, space translational symmetry of a physical system is equivalent to the
160:
156:
972:
independent translation vectors, the symmetry group is isomorphic with
767:
136:
1119:
defines a parallelogram, all with the same area, the magnitude of the
319:
980:. Different bases of translation vectors generate the same lattice
1202:
to the polynomial degree is a translation-invariant functional.
1188:
all have translational symmetries, and sometimes other kinds.
1168:) or line, respectively, fully defines the whole object.
1283:. Prometheus Books. Especially chpt. 12. Nontechnical.
1268:
List of cycles § Mathematics of waves and cycles
222:
189:
83:
40:
1279:
Stenger, Victor J. (2000) and MahouShiroUSA (2007).
269:{\displaystyle \forall \delta \ Af=A(T_{\delta }f).}
27:
Invariance of operations under geometric translation
268:
202:
119:
70:{\displaystyle f:\mathbb {R} ^{2}\to \mathbb {R} }
69:
738:Representation theory of semisimple Lie algebras
961:has an independent direction. This is in 1D a
884:
8:
1012:of the lattice). This parallelepiped is a
891:
877:
776:Particle physics and representation theory
421:
318:
303:
251:
221:
194:
188:
82:
63:
62:
53:
49:
48:
39:
1248:Translation operator (quantum mechanics)
1175:
29:
743:Representations of classical Lie groups
475:
424:
306:
183:with respect to a translation operator
34:For translational invariant functions
1045:, etc. In general in 2D, we can take
7:
596:Lie groupâLie algebra correspondence
151:of a system of equations under any
1008:the set subtends (also called the
223:
131:is an example for such a function.
25:
940:. Fundamental domains are e.g.
1103:is 1 or â1. This ensures that
791:Galilean group representations
786:Poincaré group representations
260:
244:
114:
102:
93:
87:
59:
1:
781:Lorentz group representations
748:Theorem of the highest weight
210:if the result after applying
203:{\displaystyle T_{\delta }}
179:on functions is said to be
120:{\displaystyle f(A)=f(A+t)}
1323:
733:Lie algebra representation
159:). Discrete translational
287:momentum conservation law
181:translationally invariant
728:Lie group representation
1023:E.g. in 2D, instead of
753:BorelâWeilâBott theorem
1213:translation-invariant
1181:
651:Semisimple Lie algebra
606:Adjoint representation
270:
204:
145:translational symmetry
132:
121:
71:
1179:
720:Representation theory
271:
205:
122:
72:
33:
18:Translation invariant
220:
187:
81:
38:
1297:Classical mechanics
1253:Rotational symmetry
1200:polynomial function
1198:The mapping from a
865:Table of Lie groups
706:Compact Lie algebra
163:is invariant under
1182:
1014:fundamental region
637:Affine Lie algebra
627:Simple Lie algebra
368:Special orthogonal
266:
200:
133:
117:
67:
1307:Conservation laws
1238:Periodic function
1193:Fourier transform
1031:we can also take
901:
900:
701:Split Lie algebra
664:Cartan subalgebra
526:
525:
417:Simple Lie groups
283:Noether's theorem
231:
16:(Redirected from
1314:
1281:Timeless Reality
1258:Lorentz symmetry
1228:Glide reflection
1207:Lebesgue measure
1149:wallpaper group
1102:
1076:
1060:
1044:
949:
939:
893:
886:
879:
836:Claude Chevalley
693:Complexification
536:Other Lie groups
422:
330:Classical groups
322:
304:
275:
273:
272:
267:
256:
255:
229:
215:
209:
207:
206:
201:
199:
198:
178:
170:Analogously, an
129:Lebesgue measure
126:
124:
123:
118:
76:
74:
73:
68:
66:
58:
57:
52:
21:
1322:
1321:
1317:
1316:
1315:
1313:
1312:
1311:
1287:
1286:
1276:
1243:Lattice (group)
1224:
1186:Frieze patterns
1174:
1137:lattice (group)
1094:
1062:
1046:
1036:
1018:lattice (group)
941:
908:
897:
852:
851:
850:
821:Wilhelm Killing
805:
797:
796:
795:
770:
759:
758:
757:
722:
712:
711:
710:
697:
681:
659:Dynkin diagrams
653:
643:
642:
641:
623:
601:Exponential map
590:
580:
579:
578:
559:Conformal group
538:
528:
527:
519:
511:
503:
495:
487:
468:
458:
448:
438:
419:
409:
408:
407:
388:Special unitary
332:
302:
279:Laws of physics
247:
218:
217:
211:
190:
185:
184:
174:
79:
78:
47:
36:
35:
28:
23:
22:
15:
12:
11:
5:
1320:
1318:
1310:
1309:
1304:
1299:
1289:
1288:
1285:
1284:
1275:
1272:
1271:
1270:
1265:
1260:
1255:
1250:
1245:
1240:
1235:
1230:
1223:
1220:
1219:
1218:
1203:
1196:
1189:
1173:
1170:
1006:parallelepiped
994:absolute value
986:absolute value
982:if and only if
899:
898:
896:
895:
888:
881:
873:
870:
869:
868:
867:
862:
854:
853:
849:
848:
843:
841:Harish-Chandra
838:
833:
828:
823:
818:
816:Henri Poincaré
813:
807:
806:
803:
802:
799:
798:
794:
793:
788:
783:
778:
772:
771:
766:Lie groups in
765:
764:
761:
760:
756:
755:
750:
745:
740:
735:
730:
724:
723:
718:
717:
714:
713:
709:
708:
703:
698:
696:
695:
690:
684:
682:
680:
679:
674:
668:
666:
661:
655:
654:
649:
648:
645:
644:
640:
639:
634:
629:
624:
622:
621:
616:
610:
608:
603:
598:
592:
591:
586:
585:
582:
581:
577:
576:
571:
566:
564:Diffeomorphism
561:
556:
551:
546:
540:
539:
534:
533:
530:
529:
524:
523:
522:
521:
517:
513:
509:
505:
501:
497:
493:
489:
485:
478:
477:
473:
472:
471:
470:
464:
460:
454:
450:
444:
440:
434:
427:
426:
420:
415:
414:
411:
410:
406:
405:
395:
385:
375:
365:
355:
348:Special linear
345:
338:General linear
334:
333:
328:
327:
324:
323:
315:
314:
301:
298:
294:symmetry group
265:
262:
259:
254:
250:
246:
243:
240:
237:
234:
228:
225:
197:
193:
116:
113:
110:
107:
104:
101:
98:
95:
92:
89:
86:
65:
61:
56:
51:
46:
43:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
1319:
1308:
1305:
1303:
1300:
1298:
1295:
1294:
1292:
1282:
1278:
1277:
1273:
1269:
1266:
1264:
1261:
1259:
1256:
1254:
1251:
1249:
1246:
1244:
1241:
1239:
1236:
1234:
1231:
1229:
1226:
1225:
1221:
1216:
1212:
1208:
1204:
1201:
1197:
1194:
1190:
1187:
1184:
1183:
1178:
1171:
1169:
1167:
1166:cross-section
1161:
1158:
1154:
1152:
1144:
1140:
1138:
1134:
1133:modular group
1130:
1126:
1122:
1121:cross product
1118:
1114:
1110:
1106:
1101:
1097:
1092:
1088:
1084:
1080:
1077:for integers
1075:
1072:
1068:
1065:
1059:
1056:
1052:
1049:
1043:
1039:
1034:
1030:
1026:
1021:
1019:
1015:
1011:
1007:
1004:-dimensional
1003:
999:
995:
991:
987:
983:
979:
975:
971:
966:
964:
960:
956:
953:
948:
944:
938:
935:
931:
927:
923:
919:
916:
912:
906:
894:
889:
887:
882:
880:
875:
874:
872:
871:
866:
863:
861:
858:
857:
856:
855:
847:
844:
842:
839:
837:
834:
832:
829:
827:
824:
822:
819:
817:
814:
812:
809:
808:
801:
800:
792:
789:
787:
784:
782:
779:
777:
774:
773:
769:
763:
762:
754:
751:
749:
746:
744:
741:
739:
736:
734:
731:
729:
726:
725:
721:
716:
715:
707:
704:
702:
699:
694:
691:
689:
686:
685:
683:
678:
675:
673:
670:
669:
667:
665:
662:
660:
657:
656:
652:
647:
646:
638:
635:
633:
630:
628:
625:
620:
617:
615:
612:
611:
609:
607:
604:
602:
599:
597:
594:
593:
589:
584:
583:
575:
572:
570:
567:
565:
562:
560:
557:
555:
552:
550:
547:
545:
542:
541:
537:
532:
531:
520:
514:
512:
506:
504:
498:
496:
490:
488:
482:
481:
480:
479:
474:
469:
467:
461:
459:
457:
451:
449:
447:
441:
439:
437:
431:
430:
429:
428:
423:
418:
413:
412:
403:
399:
396:
393:
389:
386:
383:
379:
376:
373:
369:
366:
363:
359:
356:
353:
349:
346:
343:
339:
336:
335:
331:
326:
325:
321:
317:
316:
313:
309:
305:
299:
297:
295:
290:
288:
284:
280:
276:
263:
257:
252:
248:
241:
238:
235:
232:
226:
214:
195:
191:
182:
177:
173:
168:
167:translation.
166:
162:
158:
154:
150:
146:
143:, continuous
142:
138:
130:
111:
108:
105:
99:
96:
90:
84:
54:
44:
41:
32:
19:
1280:
1263:Tessellation
1233:Displacement
1162:
1156:
1150:
1145:
1141:
1128:
1124:
1116:
1112:
1108:
1104:
1099:
1095:
1090:
1086:
1082:
1078:
1073:
1070:
1066:
1063:
1057:
1054:
1050:
1047:
1041:
1037:
1032:
1028:
1024:
1022:
1009:
1001:
973:
969:
967:
963:line segment
958:
954:
946:
942:
936:
933:
929:
925:
921:
917:
914:
910:
904:
902:
846:Armand Borel
831:Hermann Weyl
632:Loop algebra
614:Killing form
588:Lie algebras
465:
455:
445:
435:
401:
391:
381:
371:
361:
351:
341:
312:Lie algebras
291:
277:
212:
180:
175:
169:
144:
134:
1093:such that
998:determinant
990:determinant
826:Ălie Cartan
672:Root system
476:Exceptional
153:translation
141:mathematics
1291:Categories
1274:References
992:is 1. The
957:for which
952:hyperplane
811:Sophus Lie
804:Scientists
677:Weyl group
398:Symplectic
358:Orthogonal
308:Lie groups
149:invariance
688:Real form
574:Euclidean
425:Classical
253:δ
227:δ
224:∀
196:δ
155:(without
60:→
1302:Symmetry
1222:See also
1211:complete
1172:Examples
1010:covolume
950:for any
860:Glossary
554:Poincaré
300:Geometry
172:operator
165:discrete
161:symmetry
157:rotation
1215:measure
996:of the
988:of the
978:lattice
768:physics
549:Lorentz
378:Unitary
147:is the
137:physics
1135:, see
1089:, and
544:Circle
230:
127:. The
77:it is
1209:is a
619:Index
1205:The
1191:The
1127:and
1107:and
1061:and
1035:and
1027:and
928:} =
569:Loop
310:and
139:and
1020:.
945:+
400:Sp(
390:SU(
370:SO(
350:SL(
340:GL(
135:In
1293::
1139:.
1115:,
1100:qr
1098:â
1096:ps
1085:,
1081:,
1069:+
1053:+
1040:â
932:+
924:â
920:|
913:+
380:U(
360:O(
289:.
1217:.
1164:(
1157:p
1153:1
1151:p
1129:b
1125:a
1117:b
1113:a
1109:b
1105:a
1091:s
1087:r
1083:q
1079:p
1074:b
1071:s
1067:a
1064:r
1058:b
1055:q
1051:a
1048:p
1042:b
1038:a
1033:a
1029:b
1025:a
1002:n
974:Z
970:k
959:a
955:H
947:a
943:H
937:a
934:Z
930:p
926:Z
922:n
918:a
915:n
911:p
909:{
905:p
892:e
885:t
878:v
518:8
516:E
510:7
508:E
502:6
500:E
494:4
492:F
486:2
484:G
466:n
463:D
456:n
453:C
446:n
443:B
436:n
433:A
404:)
402:n
394:)
392:n
384:)
382:n
374:)
372:n
364:)
362:n
354:)
352:n
344:)
342:n
264:.
261:)
258:f
249:T
245:(
242:A
239:=
236:f
233:A
213:A
192:T
176:A
115:)
112:t
109:+
106:A
103:(
100:f
97:=
94:)
91:A
88:(
85:f
64:R
55:2
50:R
45::
42:f
20:)
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