629:
622:
615:
563:
556:
549:
723:
716:
709:
1595:
3358:
415:
2407:
105:
69:
2224:
1361:
702:
608:
3302:
955:
253:
207:
161:
43:
3594:
741:
542:
3330:
2594:
758:
3381:
2523:
3200:
3153:
3096:
3039:
2972:
2905:
2841:
2772:
2701:
2651:
2204:
2157:
2100:
2039:
1972:
1906:
1837:
1767:
1713:
1656:
1341:
1294:
1237:
1178:
1086:
3409:
1483:
787:
2236:
3221:
3274:
301:
403:
2347:, which is extremely similar to the cube described, with each rectangle replaced by a pentagon with one symmetry axis and 4 equal sides and 1 different side (the one corresponding to the line segment dividing the cube's face); i.e., the cube's faces bulge out at the dividing line and become narrower there. It is a subgroup of the full
427:
2342:
It is the symmetry of a cube with on each face a line segment dividing the face into two equal rectangles, such that the line segments of adjacent faces do not meet at the edge. The symmetries correspond to the even permutations of the body diagonals and the same combined with inversion. It is also
795:
one full face is a fundamental domain; other solids with the same symmetry can be obtained by adjusting the orientation of the faces, e.g. flattening selected subsets of faces to combine each subset into one face, or replacing each face by multiple faces, or a curved surface.
937:
has no subgroup of order 6. Although it is a property for the abstract group in general, it is clear from the isometry group of chiral tetrahedral symmetry: because of the chirality the subgroup would have to be
335:
of permutations of four objects, since there is exactly one such symmetry for each permutation of the vertices of the tetrahedron. The set of orientation-preserving symmetries forms a group referred to as the
454:
form 6 circles (or centrally radial lines) in the plane. Each of these 6 circles represent a mirror line in tetrahedral symmetry. The intersection of these circles meet at order 2 and 3 gyration points.
2339:. The three elements of the latter are the identity, "clockwise rotation", and "anti-clockwise rotation", corresponding to permutations of the three orthogonal 2-fold axes, preserving orientation.
896:. The three elements of the latter are the identity, "clockwise rotation", and "anti-clockwise rotation", corresponding to permutations of the three orthogonal 2-fold axes, preserving orientation.
3528:
3509:
3493:
824:
109:
904:
73:
1148:
1056:
1143:
376:
47:
2291:
is either an element of T, or one combined with inversion. Apart from these two normal subgroups, there is also a normal subgroup D
2265:. This group has the same rotation axes as T, with mirror planes through two of the orthogonal directions. The 3-fold axes are now
1398:. This group has the same rotation axes as T, but with six mirror planes, each through two 3-fold axes. The 2-fold axes are now S
3177:
3130:
3120:
3073:
3063:
3016:
3006:
2996:
2949:
2939:
2929:
2882:
2872:
2862:
2816:
2806:
2796:
2735:
2725:
2483:
2473:
2181:
2134:
2124:
2073:
2063:
2016:
2006:
1996:
1947:
1937:
1927:
1881:
1871:
1861:
1801:
1791:
1318:
1271:
1261:
1212:
1202:
1138:
1130:
1120:
1110:
1046:
1038:
1028:
1018:
692:
682:
672:
664:
654:
644:
598:
588:
578:
532:
522:
512:
504:
494:
484:
291:
281:
271:
245:
235:
225:
199:
189:
179:
143:
133:
123:
97:
87:
61:
1153:
1061:
1051:
372:
34:
2745:
2675:
2628:
2618:
2567:
2557:
2547:
2493:
1811:
1737:
1690:
1680:
1629:
1619:
1569:
1559:
1549:
628:
621:
614:
3516:
3500:
3125:
3011:
3001:
2934:
2811:
2801:
2730:
2129:
2011:
2001:
1866:
1796:
1266:
1125:
1115:
3068:
2944:
2877:
2867:
2740:
2623:
2562:
2552:
2488:
2478:
2068:
1942:
1932:
1876:
1806:
1685:
1624:
1564:
1554:
1207:
1033:
1023:
687:
677:
659:
649:
593:
583:
527:
517:
499:
489:
286:
276:
240:
230:
194:
184:
138:
128:
92:
3351:
3504:, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995,
2432:
1508:
980:
3619:
562:
555:
548:
3301:
856:
of the four 3-fold axes: e, (123), (132), (124), (142), (134), (143), (234), (243), (12)(34), (13)(24), (14)(23).
3442:
447:
3329:
3397:
3346:
3614:
722:
715:
708:
384:
317:
3374:
3295:
993:
3512:
3437:
2440:
2417:
2348:
1516:
1493:
988:
965:
257:
954:
3624:
3432:
3402:
3323:
2455:
1531:
1003:
792:
771:
750:
451:
211:
1594:
746:
327:
The group of all (not necessarily orientation preserving) symmetries is isomorphic to the group S
3357:
3598:
3575:
3524:
3505:
3489:
3290:
2427:
2266:
2240:
1962:
1503:
1076:
975:
849:
337:
3552:
3273:
2422:
2406:
2323:
1498:
970:
853:
151:
1482:
414:
104:
68:
3226:
2450:
2235:
2223:
1526:
1434:
1415:
1360:
998:
881:
860:
701:
607:
332:
252:
206:
160:
42:
740:
541:
3262:
2332:
1395:
1168:
889:
380:
321:
3593:
3578:
3557:
3540:
3380:
2522:
3608:
3318:
2445:
2351:
group (as isometry group, not just as abstract group), with 4 of the 10 3-fold axes.
1521:
17:
2593:
3408:
2344:
1227:
757:
3199:
3152:
3095:
3038:
2971:
2904:
2840:
2771:
2700:
2650:
2203:
2156:
2099:
2038:
1971:
1905:
1836:
1766:
1712:
1655:
1340:
1293:
1236:
1177:
1085:
786:
2358:
include those of T, with the two classes of 4 combined, and each with inversion:
3267:
1427:
779:
774:
format, along with the 180° edge (blue arrows) and 120° vertex (reddish arrows)
763:
313:
305:
3220:
870:
4 × rotation by 120° clockwise (seen from a vertex): (234), (143), (412), (321)
838:
3539:
Koca, Nazife; Al-Mukhaini, Aida; Koca, Mehmet; Al Qanobi, Amal (2016-12-01).
3583:
324:
of 24 including transformations that combine a reflection and a rotation.
775:
767:
2322:. It is the direct product of the normal subgroup of T (see above) with
402:
300:
2296:
1422:
is the union of T and the set obtained by combining each element of
1410:
and O are isomorphic as abstract groups: they both correspond to S
1359:
953:
735:
299:
823:. There are three orthogonal 2-fold rotation axes, like chiral
426:
3488:
2008, John H. Conway, Heidi
Burgiel, Chaim Goodman-Strauss,
3461:
880:
The rotations by 180°, together with the identity, form a
903:
is the smallest group demonstrating that the converse of
1462:
6 × reflection in a plane through two rotation axes (C
389:
308:, an example of a solid with full tetrahedral symmetry
2276:) axes, and there is a central inversion symmetry. T
833:
or 222, with in addition four 3-fold axes, centered
31:
919:|, there does not necessarily exist a subgroup of
837:the three orthogonal directions. This group is
3164:
3107:
3050:
2983:
2916:
2852:
2783:
2712:
2662:
2605:
2534:
2460:
2168:
2111:
2050:
1983:
1917:
1848:
1778:
1724:
1667:
1606:
1536:
1305:
1248:
1189:
1097:
1008:
3545:Sultan Qaboos University Journal for Science
907:is not true in general: given a finite group
873:4 × rotation by 120° counterclockwise (ditto)
8:
480:Chiral tetrahedral symmetry, T, (332), = ,
3541:"Symmetry of the Pyritohedron and Lattices"
2413:
1489:
961:
852:on 4 elements; in fact it is the group of
770:alone. These are illustrated above in the
766:can be placed in 12 distinct positions by
3556:
1478:Subgroups of achiral tetrahedral symmetry
1428:the isometries of the regular tetrahedron
782:the tetrahedron through those positions.
753:, see below, the latter is one full face
3238:
2405:
2234:
2222:
1481:
950:Subgroups of chiral tetrahedral symmetry
457:
3454:
3215:Solids with chiral tetrahedral symmetry
745:The tetrahedral rotation group T with
3235:Solids with full tetrahedral symmetry
958:Chiral tetrahedral symmetry subgroups
7:
3499:Kaleidoscopes: Selected Writings of
2402:Subgroups of pyritohedral symmetry
25:
3558:10.24200/squjs.vol21iss2pp139-149
3592:
3407:
3379:
3356:
3328:
3300:
3272:
3219:
3198:
3175:
3151:
3128:
3123:
3118:
3094:
3071:
3066:
3061:
3037:
3014:
3009:
3004:
2999:
2994:
2970:
2947:
2942:
2937:
2932:
2927:
2903:
2880:
2875:
2870:
2865:
2860:
2839:
2814:
2809:
2804:
2799:
2794:
2770:
2743:
2738:
2733:
2728:
2723:
2699:
2673:
2649:
2626:
2621:
2616:
2592:
2565:
2560:
2555:
2550:
2545:
2521:
2491:
2486:
2481:
2476:
2471:
2202:
2179:
2155:
2132:
2127:
2122:
2098:
2071:
2066:
2061:
2037:
2014:
2009:
2004:
1999:
1994:
1970:
1945:
1940:
1935:
1930:
1925:
1904:
1879:
1874:
1869:
1864:
1859:
1835:
1809:
1804:
1799:
1794:
1789:
1765:
1735:
1711:
1688:
1683:
1678:
1654:
1627:
1622:
1617:
1593:
1567:
1562:
1557:
1552:
1547:
1339:
1316:
1292:
1269:
1264:
1259:
1235:
1210:
1205:
1200:
1176:
1151:
1146:
1141:
1136:
1128:
1123:
1118:
1113:
1108:
1084:
1059:
1054:
1049:
1044:
1036:
1031:
1026:
1021:
1016:
785:
756:
739:
721:
714:
707:
700:
690:
685:
680:
675:
670:
662:
657:
652:
647:
642:
627:
620:
613:
606:
596:
591:
586:
581:
576:
561:
554:
547:
540:
530:
525:
520:
515:
510:
502:
497:
492:
487:
482:
425:
413:
401:
289:
284:
279:
274:
269:
251:
243:
238:
233:
228:
223:
205:
197:
192:
187:
182:
177:
159:
141:
136:
131:
126:
121:
103:
95:
90:
85:
67:
59:
41:
35:point groups in three dimensions
3535:, 11.5 Spherical Coxeter groups
2335:is the same as above: of type Z
821:rotational tetrahedral symmetry
636:Achiral tetrahedral symmetry, T
3597:Learning materials related to
3521:Geometries and Transformations
1:
3352:Truncated triakis tetrahedron
3224:The Icosahedron colored as a
1486:Achiral tetrahedral subgroups
381:crystallographic point groups
3395:
3372:
3344:
3316:
3288:
3260:
2393:3 × reflection in a plane (C
2386:8 × rotoreflection by 60° (S
1469:6 × rotoreflection by 90° (S
1394:, also known as the (2,3,3)
1364:The full tetrahedral group T
1356:Achiral tetrahedral symmetry
432:
394:
365:achiral tetrahedral symmetry
732:Chiral tetrahedral symmetry
3641:
2354:The conjugacy classes of T
2243:have pyritohedral symmetry
464:Stereographic projections
3345:
2421:
1497:
1426:with inversion. See also
1392:full tetrahedral symmetry
969:
635:
569:
479:
463:
460:
373:discrete point symmetries
150:
3486:The Symmetries of Things
3443:Binary tetrahedral group
2227:The pyritohedral group T
570:Pyritohedral symmetry, T
448:stereographic projection
377:symmetries on the sphere
3398:Uniform star polyhedron
3347:Near-miss Johnson solid
2372:3 × rotation by 180° (C
2365:8 × rotation by 120° (C
2231:with fundamental domain
1455:3 × rotation by 180° (C
1448:8 × rotation by 120° (C
1368:with fundamental domain
946:, but neither applies.
3533:Finite symmetry groups
2411:
2410:Pyritohedral subgroups
2244:
2232:
1487:
1369:
959:
379:). They are among the
318:orientation-preserving
316:has 12 rotational (or
309:
3375:Tetrated dodecahedron
3296:truncated tetrahedron
3231:has chiral symmetry.
2409:
2263:pyritohedral symmetry
2238:
2226:
2219:Pyritohedral symmetry
1485:
1363:
957:
369:pyritohedral symmetry
303:
48:Involutional symmetry
18:Pyritohedral symmetry
3438:Icosahedral symmetry
2349:icosahedral symmetry
2287:: every element of T
876:3 × rotation by 180°
385:cubic crystal system
338:alternating subgroup
320:) symmetries, and a
258:Icosahedral symmetry
166:Tetrahedral symmetry
3620:Rotational symmetry
3579:"Tetrahedral group"
3482:(1997), p. 295
3478:Peter R. Cromwell,
3433:Octahedral symmetry
3403:Tetrahemihexahedron
3324:triakis tetrahedron
793:tetrakis hexahedron
751:triakis tetrahedron
452:tetrakis hexahedron
392:
212:Octahedral symmetry
37:
3599:Symmetric group S4
3576:Weisstein, Eric W.
2412:
2343:the symmetry of a
2245:
2233:
1488:
1386:3m, of order 24 –
1370:
960:
905:Lagrange's theorem
747:fundamental domain
390:
375:(or equivalently,
310:
32:
3529:978-1-107-10340-5
3510:978-0-471-01003-6
3494:978-1-56881-220-5
3424:
3423:
3291:Archimedean solid
3212:
3211:
2280:is isomorphic to
2216:
2215:
1435:conjugacy classes
1353:
1352:
861:conjugacy classes
854:even permutations
850:alternating group
825:dihedral symmetry
800:
799:
729:
728:
450:the edges of the
444:
443:
298:
297:
110:Dihedral symmetry
29:3D symmetry group
16:(Redirected from
3632:
3596:
3589:
3588:
3562:
3560:
3465:
3462:Koca et al. 2016
3459:
3411:
3383:
3360:
3332:
3304:
3276:
3239:
3223:
3202:
3180:
3179:
3178:
3155:
3133:
3132:
3131:
3127:
3126:
3122:
3121:
3098:
3076:
3075:
3074:
3070:
3069:
3065:
3064:
3041:
3019:
3018:
3017:
3013:
3012:
3008:
3007:
3003:
3002:
2998:
2997:
2974:
2952:
2951:
2950:
2946:
2945:
2941:
2940:
2936:
2935:
2931:
2930:
2907:
2885:
2884:
2883:
2879:
2878:
2874:
2873:
2869:
2868:
2864:
2863:
2843:
2827:
2819:
2818:
2817:
2813:
2812:
2808:
2807:
2803:
2802:
2798:
2797:
2774:
2748:
2747:
2746:
2742:
2741:
2737:
2736:
2732:
2731:
2727:
2726:
2703:
2686:
2678:
2677:
2676:
2653:
2631:
2630:
2629:
2625:
2624:
2620:
2619:
2596:
2570:
2569:
2568:
2564:
2563:
2559:
2558:
2554:
2553:
2549:
2548:
2525:
2505:
2496:
2495:
2494:
2490:
2489:
2485:
2484:
2480:
2479:
2475:
2474:
2414:
2321:
2286:
2275:
2261:, of order 24 –
2260:
2206:
2184:
2183:
2182:
2159:
2137:
2136:
2135:
2131:
2130:
2126:
2125:
2102:
2076:
2075:
2074:
2070:
2069:
2065:
2064:
2041:
2019:
2018:
2017:
2013:
2012:
2008:
2007:
2003:
2002:
1998:
1997:
1974:
1950:
1949:
1948:
1944:
1943:
1939:
1938:
1934:
1933:
1929:
1928:
1908:
1892:
1884:
1883:
1882:
1878:
1877:
1873:
1872:
1868:
1867:
1863:
1862:
1839:
1822:
1814:
1813:
1812:
1808:
1807:
1803:
1802:
1798:
1797:
1793:
1792:
1769:
1748:
1740:
1739:
1738:
1715:
1693:
1692:
1691:
1687:
1686:
1682:
1681:
1658:
1632:
1631:
1630:
1626:
1625:
1621:
1620:
1597:
1580:
1572:
1571:
1570:
1566:
1565:
1561:
1560:
1556:
1555:
1551:
1550:
1490:
1425:
1405:
1385:
1343:
1321:
1320:
1319:
1296:
1274:
1273:
1272:
1268:
1267:
1263:
1262:
1239:
1215:
1214:
1213:
1209:
1208:
1204:
1203:
1180:
1156:
1155:
1154:
1150:
1149:
1145:
1144:
1140:
1139:
1133:
1132:
1131:
1127:
1126:
1122:
1121:
1117:
1116:
1112:
1111:
1088:
1064:
1063:
1062:
1058:
1057:
1053:
1052:
1048:
1047:
1041:
1040:
1039:
1035:
1034:
1030:
1029:
1025:
1024:
1020:
1019:
962:
936:
815:, of order 12 –
789:
760:
743:
736:
725:
718:
711:
704:
695:
694:
693:
689:
688:
684:
683:
679:
678:
674:
673:
667:
666:
665:
661:
660:
656:
655:
651:
650:
646:
645:
631:
624:
617:
610:
601:
600:
599:
595:
594:
590:
589:
585:
584:
580:
579:
565:
558:
551:
544:
535:
534:
533:
529:
528:
524:
523:
519:
518:
514:
513:
507:
506:
505:
501:
500:
496:
495:
491:
490:
486:
485:
458:
429:
417:
405:
393:
294:
293:
292:
288:
287:
283:
282:
278:
277:
273:
272:
255:
248:
247:
246:
242:
241:
237:
236:
232:
231:
227:
226:
209:
202:
201:
200:
196:
195:
191:
190:
186:
185:
181:
180:
163:
152:Polyhedral group
146:
145:
144:
140:
139:
135:
134:
130:
129:
125:
124:
107:
100:
99:
98:
94:
93:
89:
88:
71:
64:
63:
62:
45:
38:
21:
3640:
3639:
3635:
3634:
3633:
3631:
3630:
3629:
3605:
3604:
3574:
3573:
3570:
3565:
3538:
3474:
3469:
3468:
3460:
3456:
3451:
3429:
3237:
3217:
3195:
3176:
3174:
3169:
3148:
3129:
3124:
3119:
3117:
3112:
3091:
3072:
3067:
3062:
3060:
3055:
3034:
3015:
3010:
3005:
3000:
2995:
2993:
2988:
2967:
2948:
2943:
2938:
2933:
2928:
2926:
2921:
2900:
2881:
2876:
2871:
2866:
2861:
2859:
2836:
2825:
2815:
2810:
2805:
2800:
2795:
2793:
2788:
2767:
2763:
2744:
2739:
2734:
2729:
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2199:
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2015:
2010:
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1995:
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1966:
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1472:
1465:
1458:
1451:
1440:
1423:
1421:
1418:on 4 objects. T
1416:symmetric group
1413:
1409:
1403:
1401:
1383:
1376:
1367:
1358:
1336:
1317:
1315:
1310:
1289:
1270:
1265:
1260:
1258:
1253:
1231:
1211:
1206:
1201:
1199:
1194:
1172:
1152:
1147:
1142:
1137:
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1129:
1124:
1119:
1114:
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1055:
1050:
1045:
1043:
1037:
1032:
1027:
1022:
1017:
1015:
952:
945:
941:
935:
928:
902:
895:
887:
882:normal subgroup
847:
832:
790:
761:
744:
734:
691:
686:
681:
676:
671:
669:
663:
658:
653:
648:
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640:, (*332), = ,
639:
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333:symmetric group
330:
290:
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142:
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132:
127:
122:
120:
118:
116:
112:
108:
96:
91:
86:
84:
82:
80:
76:
74:Cyclic symmetry
72:
60:
58:
56:
54:
50:
46:
30:
23:
22:
15:
12:
11:
5:
3638:
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3601:at Wikiversity
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3568:External links
3566:
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3501:H.S.M. Coxeter
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1396:triangle group
1374:
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991:
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983:
978:
973:
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951:
948:
943:
939:
933:
911:and a divisor
900:
893:
890:quotient group
885:
878:
877:
874:
871:
868:
845:
830:
798:
797:
783:
754:
733:
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435:
431:
430:
421:
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409:
406:
397:
391:Gyration axes
353:
350:
345:
341:
328:
322:symmetry order
296:
295:
262:
249:
216:
203:
170:
156:
155:
148:
147:
114:
101:
78:
65:
52:
28:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3637:
3626:
3623:
3621:
3618:
3616:
3615:Finite groups
3613:
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3591:
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3585:
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3577:
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3522:
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3515:
3513:
3511:
3507:
3503:
3502:
3497:
3495:
3491:
3487:
3484:
3481:
3477:
3476:
3471:
3463:
3458:
3455:
3448:
3444:
3441:
3439:
3436:
3434:
3431:
3430:
3426:
3419:
3416:
3413:
3410:
3406:
3404:
3401:
3399:
3396:
3391:
3388:
3385:
3382:
3378:
3376:
3373:
3368:
3365:
3362:
3359:
3355:
3353:
3350:
3348:
3340:
3337:
3334:
3331:
3327:
3325:
3322:
3320:
3319:Catalan solid
3317:
3312:
3309:
3306:
3303:
3299:
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3278:
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3141:
3138:
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3116:
3114:
3108:
3103:
3100:
3097:
3093:
3087:
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2990:
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2501:
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2298:
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2256:
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2237:
2225:
2218:
2211:
2208:
2205:
2201:
2195:
2192:
2189:
2186:
2177:
2175:
2169:
2164:
2161:
2158:
2154:
2148:
2145:
2142:
2139:
2120:
2118:
2112:
2107:
2104:
2101:
2097:
2087:
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2081:
2078:
2059:
2057:
2051:
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2036:
2030:
2027:
2024:
2021:
1992:
1990:
1984:
1979:
1976:
1973:
1969:
1967:
1961:
1958:
1955:
1952:
1923:
1921:
1918:
1913:
1910:
1907:
1903:
1897:
1894:
1889:
1886:
1857:
1855:
1849:
1844:
1841:
1838:
1834:
1828:
1825:
1819:
1816:
1787:
1785:
1779:
1774:
1771:
1768:
1764:
1754:
1751:
1745:
1742:
1733:
1731:
1725:
1720:
1717:
1714:
1710:
1704:
1701:
1698:
1695:
1676:
1674:
1668:
1663:
1660:
1657:
1653:
1643:
1640:
1637:
1634:
1615:
1613:
1607:
1602:
1599:
1596:
1592:
1586:
1583:
1577:
1574:
1545:
1543:
1537:
1533:
1530:
1528:
1525:
1523:
1520:
1518:
1515:
1512:
1510:
1507:
1505:
1502:
1500:
1495:
1492:
1491:
1484:
1477:
1468:
1461:
1454:
1447:
1444:
1443:
1442:
1436:
1431:
1429:
1417:
1397:
1393:
1389:
1381:
1377:
1362:
1355:
1348:
1345:
1342:
1338:
1332:
1329:
1326:
1323:
1314:
1312:
1306:
1301:
1298:
1295:
1291:
1285:
1282:
1279:
1276:
1257:
1255:
1249:
1244:
1241:
1238:
1234:
1232:
1226:
1223:
1220:
1217:
1198:
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1190:
1185:
1182:
1179:
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1173:
1167:
1164:
1161:
1158:
1106:
1104:
1098:
1093:
1090:
1087:
1083:
1081:
1075:
1072:
1069:
1066:
1014:
1012:
1009:
1005:
1002:
1000:
997:
995:
992:
990:
987:
984:
982:
979:
977:
974:
972:
967:
964:
963:
956:
949:
947:
931:
926:
922:
918:
914:
910:
906:
897:
891:
883:
875:
872:
869:
866:
865:
864:
862:
857:
855:
851:
844:
840:
836:
829:
826:
822:
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781:
777:
773:
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748:
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731:
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634:
630:
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619:
616:
612:
609:
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568:
564:
560:
557:
553:
550:
546:
543:
539:
538:
478:
474:
471:
468:
467:
459:
456:
453:
449:
439:
436:
433:
428:
419:
416:
407:
404:
395:
388:
386:
382:
378:
374:
370:
366:
362:
358:
351:
349:
339:
334:
325:
323:
319:
315:
307:
302:
259:
254:
250:
213:
208:
204:
167:
162:
158:
157:
153:
149:
111:
106:
102:
75:
70:
66:
49:
44:
40:
39:
36:
27:
19:
3582:
3548:
3544:
3532:
3531:Chapter 11:
3520:
3517:N.W. Johnson
3498:
3485:
3479:
3457:
3225:
3218:
2379:inversion (S
2353:
2345:pyritohedron
2341:
2326:
2262:
2254:
2247:
2246:
1432:
1391:
1387:
1379:
1372:
1371:
929:
927:: the group
924:
920:
916:
912:
908:
898:
879:
858:
842:
834:
827:
820:
816:
812:
808:
803:
802:
801:
445:
368:
364:
360:
356:
355:
326:
311:
165:
26:
3268:tetrahedron
3229:tetrahedron
2437:Generators
2299:), of type
2295:(that of a
1513:Generators
985:Generators
923:with order
884:of type Dih
772:cycle graph
764:tetrahedron
574:, (3*2), ,
461:Orthogonal
314:tetrahedron
306:tetrahedron
154:, , (*n32)
3625:Tetrahedra
3609:Categories
3551:(2): 139.
3472:References
2241:volleyball
863:of T are:
839:isomorphic
749:; for the
312:A regular
304:A regular
3584:MathWorld
3523:, (2018)
3480:Polyhedra
3449:Citations
3257:Vertices
2441:Structure
1517:Structure
1406:) axes. T
989:Structure
892:of type Z
776:rotations
33:Selected
3427:See also
3248:Picture
2362:identity
1445:identity
867:identity
768:rotation
446:Seen in
265:, (*532)
219:, (*432)
173:, (*332)
117:, (*n22)
2423:Coxeter
2331:. The
2257:, or m
1499:Coxeter
1388:achiral
971:Coxeter
888:, with
835:between
811:, , or
791:In the
780:permute
475:2-fold
472:3-fold
469:4-fold
383:of the
352:Details
81:, (*nn)
3527:
3508:
3492:
3254:Edges
3251:Faces
3242:Class
2418:Schoe.
2297:cuboid
1494:Schoe.
1414:, the
1382:, or
966:Schoe.
848:, the
817:chiral
371:) are
357:Chiral
331:, the
3245:Name
2456:Index
2451:Order
2282:T × Z
1532:Index
1527:Order
1441:are:
1424:O \ T
1004:Index
999:Order
778:that
55:, (*)
3525:ISBN
3506:ISBN
3490:ISBN
3227:snub
2687:or m
2573:*222
2428:Orb.
1749:or m
1575:*332
1504:Orb.
1437:of T
1433:The
1380:*332
976:Orb.
942:or D
915:of |
859:The
367:and
363:(or
361:full
359:and
344:of S
3553:doi
3392:28
3389:54
3386:28
3369:28
3366:42
3363:16
3313:12
3208:24
3161:12
3025:222
3022:222
2955:322
2888:332
2849:12
2754:2/m
2709:12
2637:mm2
2634:*22
2576:mmm
2499:3*2
2446:Cyc
2433:H-M
2317:× Z
2313:× Z
2309:= Z
2305:× Z
2301:Dih
2255:3*2
2212:24
2165:12
2092:= A
2025:222
2022:222
1953:332
1817:2*2
1775:12
1759:= D
1699:mm2
1696:*22
1635:*33
1522:Cyc
1509:H-M
1390:or
1349:12
1162:222
1159:222
1067:332
994:Cyc
981:H-M
932:= A
841:to
819:or
809:332
3611::
3581:.
3549:21
3547:.
3543:.
3519::
3420:6
3417:12
3341:8
3338:18
3335:12
3310:18
3285:4
3183:11
3136:22
3104:8
3079:33
3047:6
2980:4
2913:2
2910:12
2891:23
2780:6
2764:×D
2751:2*
2716:2h
2659:6
2609:2v
2602:3
2586:×D
2538:2h
2531:1
2528:24
2515:×Z
2513:4
2293:2h
2253:,
2187:11
2140:22
2108:8
2079:33
2047:6
1980:2
1977:12
1956:23
1914:6
1887:2×
1845:3
1823:2m
1782:2d
1721:6
1671:2v
1664:4
1648:=S
1638:3m
1610:3v
1603:1
1600:24
1581:3m
1430:.
1378:,
1324:11
1302:6
1277:22
1245:4
1218:33
1186:3
1134:=
1094:1
1091:12
1070:23
1042:=
813:23
807:,
762:A
668:=
508:=
440:3
387:.
348:.
267:=
221:=
175:=
119:=
115:nh
83:=
79:nv
57:=
3587:.
3561:.
3555::
3464:.
3414:7
3307:8
3282:6
3279:4
3205:1
3194:1
3192:Z
3189:1
3186:1
3168:1
3166:C
3158:2
3147:2
3145:Z
3142:1
3139:2
3111:2
3109:C
3101:3
3090:3
3088:Z
3085:1
3082:3
3054:3
3052:C
3044:4
3033:8
3031:D
3028:3
2987:2
2985:D
2977:6
2966:6
2964:D
2961:2
2958:3
2920:3
2918:D
2899:4
2897:A
2894:2
2854:T
2846:2
2835:2
2833:Z
2830:1
2826:1
2822:×
2787:2
2785:S
2777:4
2766:2
2762:2
2760:Z
2757:2
2714:C
2706:2
2695:2
2693:D
2690:1
2685:2
2681:*
2666:s
2664:C
2656:4
2645:4
2643:D
2640:2
2607:C
2599:8
2588:2
2584:4
2582:D
2579:3
2536:D
2517:2
2511:A
2508:2
2504:3
2502:m
2464:h
2462:T
2397:)
2395:s
2390:)
2388:6
2383:)
2381:2
2376:)
2374:2
2369:)
2367:3
2356:h
2337:3
2327:i
2324:C
2319:2
2315:2
2311:2
2307:2
2303:2
2289:h
2284:2
2278:h
2274:3
2272:(
2269:6
2267:S
2259:3
2250:h
2248:T
2229:h
2209:1
2198:1
2196:Z
2193:1
2190:1
2172:1
2170:C
2162:2
2151:2
2149:Z
2146:1
2143:2
2115:2
2113:C
2105:3
2094:3
2090:3
2088:Z
2085:1
2082:3
2054:3
2052:C
2044:4
2033:4
2031:D
2028:2
1987:2
1985:D
1965:4
1963:A
1959:2
1919:T
1911:4
1900:4
1898:Z
1895:1
1891:4
1852:4
1850:C
1842:8
1831:8
1829:D
1826:2
1821:4
1780:D
1772:2
1761:2
1757:2
1755:Z
1752:1
1747:2
1743:*
1728:s
1726:C
1718:4
1707:4
1705:D
1702:2
1669:C
1661:6
1650:3
1646:6
1644:D
1641:2
1608:C
1589:4
1587:S
1584:3
1579:4
1540:d
1538:T
1473:)
1471:4
1466:)
1464:s
1459:)
1457:2
1452:)
1450:3
1439:d
1420:d
1412:4
1408:d
1404:4
1402:(
1400:4
1384:4
1375:d
1373:T
1366:d
1346:1
1335:1
1333:Z
1330:1
1327:1
1309:1
1307:C
1299:2
1288:2
1286:Z
1283:1
1280:2
1252:2
1250:C
1242:3
1230:3
1228:Z
1224:1
1221:3
1193:3
1191:C
1183:4
1171:4
1169:D
1165:3
1101:2
1099:D
1079:4
1077:A
1073:2
1010:T
944:3
940:6
938:C
934:4
930:G
925:d
921:G
917:G
913:d
909:G
901:4
899:A
894:3
886:2
846:4
843:A
831:2
828:D
804:T
638:d
572:h
437:2
434:2
422:2
420:C
410:3
408:C
398:3
396:C
346:4
342:4
340:A
329:4
263:h
261:I
217:h
215:O
171:d
169:T
113:D
77:C
53:s
51:C
20:)
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