694:
3181:
886:
984:
877:
868:
859:
1033:
1015:
975:
966:
957:
939:
1024:
948:
841:
2957:
850:
3302:
3199:
2131:
3256:
710:
678:
2341:
2293:
3289:
3278:
3267:
2348:
2054:
2279:
2198:
2117:
2040:
1963:
1894:
1817:
3172:
3163:
3152:
3141:
3128:
3119:
3110:
3101:
3090:
3074:
3063:
3054:
3045:
3034:
3021:
3010:
3001:
2990:
2979:
2966:
2946:
2935:
2365:
3344:
3333:
3322:
3313:
3243:
3232:
3221:
3210:
1977:
2372:
2212:
1908:
1831:
2286:
2205:
2124:
1970:
1901:
734:
object without translational symmetry can be described by the 3D symmetry type by adding a third dimension to the object which does not add or spoil symmetry. For example, for a 2D image we can consider a piece of carton with that image displayed on one side; the shape of the carton should be such
2047:
1824:
638:
If the sum of the feature values is 2, the order is infinite, i.e., the notation represents a wallpaper group or a frieze group. Indeed, Conway's "Magic
Theorem" indicates that the 17 wallpaper groups are exactly those with the sum of the feature values equal to 2. Otherwise, the order is 2
423:
indicates infinite rotational symmetry around a line; it can only occur for bold face groups. By abuse of language, we might say that such a group is a subgroup of symmetries of the
Euclidean plane with only one independent translation. The
766:
image can be drawn horizontally on a piece of carton, with a provision to avoid additional symmetry with respect to the line of the image, e.g. by drawing a horizontal bar under the image. Thus the discrete
334:
represents a group of symmetries of
Euclidean 3-space. A string not written in boldface represents a group of symmetries of the Euclidean plane, which is assumed to contain two independent translations.
3852:
46:
in two-dimensional spaces of constant curvature. The advantage of the notation is that it describes these groups in a way which indicates many of the groups' properties: in particular, it follows
2221:
This group is generated by a translation and the reflection in the horizontal axis. The glide reflection here arises as the composition of translation and horizontal reflection
601:
556:
2162:
2152:
2095:
2085:
2018:
2008:
1998:
1872:
1862:
1852:
1793:
1783:
242:
2302:
This group requires three generators, with one generating set consisting of a translation, the reflection in the horizontal axis and a reflection across a vertical axis.
2238:
2157:
2080:
2003:
1934:
1857:
1788:
2253:
2243:
2233:
2172:
2075:
1939:
1929:
2090:
2013:
1867:
2248:
2167:
623:
421:
398:
316:
270:
164:
133:
94:
2316:
in yellow, with reflection lines in blue, glide reflection lines in dashed green, translation normals in red, and 2-fold gyration points as small green squares.
2140:
The translations here arise from the glide reflections, so this group is generated by a glide reflection and either a rotation or a vertical reflection.
1986:
The group is the same as the non-trivial group in the one-dimensional case; it is generated by a translation and a reflection in the vertical axis.
3855:
Software for visualizing two-dimensional tilings of the plane, sphere and hyperbolic plane, and editing their symmetry groups in orbifold notation
747:(•) is added on one- and two-dimensional groups to imply the existence of a fixed point. (In three dimensions these groups exist in an n-fold
3714:
3750:
J. H. Conway, D. H. Huson. The
Orbifold Notation for Two-Dimensional Groups. Structural Chemistry, 13 (3-4): 247–257, August 2002.
3838:
3780:
1917:
This group is singly generated, by a glide reflection, with translations being obtained by combining two glide reflections.
3743:
John H. Conway, Olaf
Delgado Friedrichs, Daniel H. Huson, and William P. Thurston. On Three-dimensional Space Groups.
1044:
768:
62:
3757:, Proceedings of the L.M.S. Durham Symposium, July 5–15, Durham, UK, 1990; London Math. Soc. Lecture Notes Series
2309:
Schönflies's point group notation is extended here as infinite cases of the equivalent dihedral points symmetries
3878:
3873:
1722:
1840:
This group is singly generated, by a translation by the smallest distance over which the pattern is periodic.
194:
All translations which occur are assumed to form a discrete subgroup of the group symmetries being described.
3845:
in
Minneapolis, with John Conway, Peter Doyle, Jane Gilman and Bill Thurston, on June 17–28, 1991. See also
2399:
1070:
774:
Another way of constructing a 3D object from a 1D or 2D object for describing the symmetry is taking the
295:
because it topologically represents a torus (1-handle) closed surface. Patterns repeat by two translation.
3883:
1065:
3753:
J. H. Conway (1992). "The
Orbifold Notation for Surface Groups". In: M. W. Liebeck and J. Saxl (eds.),
566:
526:
174:
The following types of
Euclidean transformation can occur in a group described by orbifold notation:
3842:
209:
197:
Each group is denoted in orbifold notation by a finite string made up from the following symbols:
1950:
3735:
Symmetries of Things, Chapter 18, More on
Hyperbolic groups, Enumerating hyperbolic groups, p239
3786:
Hughes, Sam (2022), "Cohomology of
Fuchsian groups and non-Euclidean crystallographic groups",
3717:
Mathematician John Conway created names that relate to footsteps for each of the frieze groups.
2917:
3868:
3768:
3677:
1804:
608:
406:
383:
301:
255:
3805:
1750:
509:
142:
111:
72:
693:
8:
744:
136:
3809:
735:
that it does not spoil the symmetry, or it can be imagined to be infinite. Thus we have
3821:
3795:
3764:
3682:
2313:
1765:
39:
3846:
3825:
3776:
3293:
2138:(TRVG) Vertical reflection lines, Glide reflections, Translations and 180° Rotations:
775:
716:
885:
3813:
3657:*23∞, *2 4 12, *266, 6*2, *336, 3*6, *344, 4*3, *2223, 2*23, 2 3 12, 246, 334
3371:
3282:
3180:
2394:
1745:
1060:
983:
47:
35:
2300:(TRHVG) Horizontal and Vertical reflection lines, Translations and 180° Rotations:
876:
867:
858:
3271:
2406:
2382:
1032:
1014:
974:
965:
956:
377:
which rotates around a kaleidoscopic point and reflects through a line (or plane)
105:
101:
55:
43:
1023:
840:
3817:
3260:
2956:
2921:
938:
849:
355:
947:
16:
Notation for 2-dimensional spherical, euclidean and hyperbolic symmetry groups
3862:
3348:
3337:
3326:
3306:
3247:
3236:
3225:
3214:
3203:
763:
731:
497:
3185:
3156:
3145:
3132:
3094:
3067:
3038:
3025:
3014:
2994:
2983:
2970:
2950:
2939:
1740:
1733:
635:
Subtracting the sum of these values from 2 gives the Euler characteristic.
516:
can be read from its Conway symbol, as follows. Each feature has a value:
425:
97:
3356:
A first few hyperbolic groups, ordered by their Euler characteristic are:
3301:
3198:
3686:
2410:
435:
indicates that there are precisely two linearly independent translations.
326:
where a pattern repeats as a mirror image without crossing a mirror line.
3255:
2130:
3288:
3277:
3266:
2340:
2292:
2285:
2204:
2123:
2053:
1969:
1900:
677:
492:
if its symmetry group contains no reflections; otherwise it is called
684:
2347:
2046:
1823:
709:
3800:
727:
700:
513:
362:
347:
331:
323:
276:
248:
51:
25:
20:
2278:
2197:
2116:
2039:
1962:
1893:
1816:
3171:
3162:
3151:
3140:
3127:
3118:
3109:
3100:
3089:
3073:
3062:
3053:
3044:
3033:
3020:
3009:
3000:
2989:
2978:
2965:
2945:
2934:
373:
to the right of an asterisk indicates a transformation of order 2
203:
3343:
3332:
3321:
3312:
3242:
3231:
3220:
3209:
1976:
778:
of the object and an asymmetric 2D or 1D object, respectively.
66:
2219:(THG) Translations, Horizontal reflections, Glide reflections:
2211:
1907:
2063:
The group is generated by a translation and a 180° rotation.
1830:
748:
2371:
2364:
190:
glide-reflection, i.e. reflection followed by translation.
3761:. Cambridge University Press, Cambridge. pp. 438–447
664:
This is because 1-fold rotation is the "empty" rotation.
503:
318:(an open circle in older documents), which is called a
287:(a solid circle in older documents), which is called a
338:
Each symbol corresponds to a distinct transformation:
611:
569:
529:
409:
386:
304:
258:
212:
145:
114:
75:
2326:Fundamental domains of Euclidean reflective groups
792:Fundamental domains of reflective 3D point groups
617:
595:
550:
415:
392:
310:
264:
236:
158:
127:
88:
61:Groups representable in this notation include the
703:has symmetry *5•, the whole image with arrows 5•.
500:in the chiral case and non-orientable otherwise.
3860:
1984:(TV) Vertical reflection lines and Translations:
169:
34:) is a system, invented by the mathematician
3685:- an extension of orbifold notation for 3d
3518:*2 3 19, *2 3 20, *2 3 21, *2 3 22, *2 3 23
483:
3726:Symmetries of Things, Appendix A, page 416
3705:Symmetries of Things, Appendix A, page 416
187:infinite rotation around a line in 3-space
3799:
667:
1915:(TG) Glide-reflections and Translations:
781:
523:without or before an asterisk counts as
346:to the left of an asterisk indicates a
184:rotation of finite order around a point
3861:
3785:
504:The Euler characteristic and the order
3745:Contributions to Algebra and Geometry
2061:(TR) Translations and 180° Rotations:
771:are *•, *1•, ∞• and *∞•.
647:The following groups are isomorphic:
639:divided by the Euler characteristic.
178:reflection through a line (or plane)
3192:Example higher polygons (*pqrs...)
2911:
2320:
448:if it is not one of the following:
54:obtained by taking the quotient of
13:
3755:Groups, Combinatorics and Geometry
3577:*2 3 36 ... *2 3 70, *249, *2 4 10
1716:
410:
259:
58:by the group under consideration.
14:
3895:
3832:
3083:Example general triangles (*pqr)
1045:List of spherical symmetry groups
719:has 5 fold rotation symmetry, 5•.
596:{\displaystyle {\frac {n-1}{2n}}}
439:
3540:*2 3 30, *256, *335, 3*5, 2 3 10
3342:
3331:
3320:
3311:
3300:
3287:
3276:
3265:
3254:
3241:
3230:
3219:
3208:
3197:
3179:
3170:
3161:
3150:
3139:
3126:
3117:
3108:
3099:
3088:
3072:
3061:
3052:
3043:
3032:
3019:
3008:
2999:
2988:
2977:
2964:
2955:
2944:
2933:
2370:
2363:
2346:
2339:
2291:
2284:
2277:
2251:
2246:
2241:
2236:
2231:
2210:
2203:
2196:
2170:
2165:
2160:
2155:
2150:
2129:
2122:
2115:
2093:
2088:
2083:
2078:
2073:
2052:
2045:
2038:
2016:
2011:
2006:
2001:
1996:
1975:
1968:
1961:
1937:
1932:
1927:
1906:
1899:
1892:
1870:
1865:
1860:
1855:
1850:
1829:
1822:
1815:
1791:
1786:
1781:
1727:
1031:
1022:
1013:
982:
973:
964:
955:
946:
937:
884:
875:
866:
857:
848:
839:
769:symmetry groups in one dimension
751:orbifold and are represented as
708:
692:
676:
551:{\displaystyle {\frac {n-1}{n}}}
496:. The corresponding orbifold is
2928:Example right triangles (*2pq)
642:
3843:"Geometry and the Imagination"
3839:A field guide to the orbifolds
3729:
3720:
3708:
3699:
1723:List of planar symmetry groups
135:), and their analogues on the
1:
3692:
3442:*2 3 12, *246, *334, 3*4, 238
444:An orbifold symbol is called
322:and represents a topological
2290:
2209:
2128:
2051:
1974:
1905:
1828:
1212:Dihedral and cyclic groups:
786:
563:after an asterisk counts as
400:indicates a glide reflection
237:{\displaystyle 1,2,3,\dots }
42:, for representing types of
7:
3671:
3360:Hyperbolic symmetry groups
10:
3900:
3818:10.1007/s00229-022-01369-z
1720:
1050:Spherical symmetry groups
1042:
366:, * indicates a reflection
170:Definition of the notation
3664:
3191:
3082:
2927:
1762:
1375:
1211:
1079:
1030:
1021:
1012:
1004:
998:
992:
3773:The Symmetries of Things
687:would have *6• symmetry,
484:Chirality and achirality
3788:Manuscripta Mathematica
3464:*2 3 15, *255, 5*2, 245
618:{\displaystyle \times }
431:the exceptional symbol
416:{\displaystyle \infty }
393:{\displaystyle \times }
311:{\displaystyle \times }
265:{\displaystyle \infty }
181:translation by a vector
3747:, 42(2):475-507, 2001.
2920:of fundamental domain
2312:The diagram shows one
1838:(T) Translations only:
668:Two-dimensional groups
619:
597:
552:
417:
394:
312:
266:
238:
160:
129:
90:
3879:Mathematical notation
3874:Generalized manifolds
3841:(Notes from class on
3769:Chaim Goodman-Strauss
3678:Mutation of orbifolds
3646:*2 3 132, *2 4 11 ...
782:Correspondence tables
620:
598:
553:
418:
395:
313:
267:
239:
161:
159:{\displaystyle H^{2}}
130:
128:{\displaystyle E^{2}}
91:
89:{\displaystyle S^{2}}
3660:, , , , , , , , ,
609:
567:
527:
510:Euler characteristic
407:
384:
330:A string written in
302:
256:
210:
143:
112:
73:
3810:2019arXiv191000519H
3361:
2924:
2918:Poincaré disk model
2385:
2327:
1736:
1051:
793:
3884:John Horton Conway
3683:Fibrifold notation
3359:
2916:
2380:
2325:
2314:fundamental domain
1732:
1080:Polyhedral groups
1049:
791:
615:
593:
548:
428:occur in this way.
413:
390:
308:
262:
234:
156:
125:
86:
50:in describing the
32:orbifold signature
3781:978-1-56881-220-5
3767:, Heidi Burgiel,
3669:
3668:
3354:
3353:
3167:*3∞∞
3078:*2∞∞
2909:
2908:
2378:
2377:
2306:
2305:
1714:
1713:
1041:
1040:
776:Cartesian product
717:Flag of Hong Kong
591:
546:
3891:
3828:
3803:
3794:(3–4): 659–676,
3736:
3733:
3727:
3724:
3718:
3712:
3706:
3703:
3643:
3642:
3638:
3635:
3620:
3619:
3615:
3612:
3597:
3596:
3592:
3589:
3574:
3573:
3569:
3566:
3560:
3559:
3555:
3552:
3494:
3493:
3489:
3486:
3453:*2 3 13, *2 3 14
3362:
3358:
3346:
3335:
3324:
3315:
3304:
3291:
3280:
3269:
3258:
3245:
3234:
3223:
3212:
3201:
3183:
3174:
3165:
3154:
3143:
3130:
3121:
3112:
3103:
3092:
3076:
3065:
3056:
3047:
3036:
3023:
3012:
3003:
2992:
2981:
2968:
2959:
2948:
2937:
2925:
2915:
2912:Hyperbolic plane
2386:
2383:wallpaper groups
2379:
2374:
2367:
2350:
2343:
2328:
2324:
2321:Wallpaper groups
2295:
2288:
2281:
2256:
2255:
2254:
2250:
2249:
2245:
2244:
2240:
2239:
2235:
2234:
2214:
2207:
2200:
2175:
2174:
2173:
2169:
2168:
2164:
2163:
2159:
2158:
2154:
2153:
2133:
2126:
2119:
2098:
2097:
2096:
2092:
2091:
2087:
2086:
2082:
2081:
2077:
2076:
2056:
2049:
2042:
2021:
2020:
2019:
2015:
2014:
2010:
2009:
2005:
2004:
2000:
1999:
1979:
1972:
1965:
1942:
1941:
1940:
1936:
1935:
1931:
1930:
1910:
1903:
1896:
1875:
1874:
1873:
1869:
1868:
1864:
1863:
1859:
1858:
1854:
1853:
1833:
1826:
1819:
1796:
1795:
1794:
1790:
1789:
1785:
1784:
1737:
1731:
1688:
1684:
1658:
1585:
1581:
1554:
1509:
1505:
1483:
1422:
1418:
1395:
1349:
1345:
1323:
1262:
1258:
1235:
1168:
1052:
1048:
1035:
1026:
1017:
986:
977:
968:
959:
950:
941:
888:
879:
870:
861:
852:
843:
794:
790:
712:
696:
680:
624:
622:
621:
616:
602:
600:
599:
594:
592:
590:
582:
571:
557:
555:
554:
549:
547:
542:
531:
422:
420:
419:
414:
399:
397:
396:
391:
317:
315:
314:
309:
271:
269:
268:
263:
243:
241:
240:
235:
165:
163:
162:
157:
155:
154:
137:hyperbolic plane
134:
132:
131:
126:
124:
123:
102:wallpaper groups
95:
93:
92:
87:
85:
84:
48:William Thurston
38:and promoted by
36:William Thurston
3899:
3898:
3894:
3893:
3892:
3890:
3889:
3888:
3859:
3858:
3835:
3740:
3739:
3734:
3730:
3725:
3721:
3715:Frieze Patterns
3713:
3709:
3704:
3700:
3695:
3674:
3640:
3636:
3633:
3631:
3617:
3613:
3610:
3608:
3600:*2 3 66, 2 3 11
3594:
3590:
3587:
3585:
3571:
3567:
3564:
3562:
3557:
3553:
3550:
3548:
3491:
3487:
3484:
3482:
3347:
3336:
3325:
3316:
3305:
3292:
3281:
3270:
3259:
3246:
3235:
3224:
3213:
3202:
3184:
3175:
3166:
3155:
3144:
3131:
3122:
3113:
3104:
3093:
3077:
3066:
3057:
3048:
3037:
3024:
3013:
3004:
2993:
2982:
2969:
2960:
2949:
2938:
2914:
2905:
2899:
2893:
2877:
2870:
2864:
2848:
2841:
2835:
2819:
2812:
2806:
2790:
2784:
2778:
2762:
2755:
2749:
2733:
2726:
2720:
2704:
2697:
2691:
2675:
2668:
2662:
2646:
2640:
2634:
2618:
2612:
2606:
2590:
2584:
2578:
2562:
2556:
2550:
2534:
2528:
2522:
2506:
2500:
2494:
2478:
2472:
2466:
2450:
2444:
2438:
2421:
2416:
2409:
2401:
2390:
2323:
2301:
2296:
2271:
2267:
2263:
2262:
2252:
2247:
2242:
2237:
2232:
2230:
2229:
2220:
2215:
2190:
2186:
2182:
2181:
2171:
2166:
2161:
2156:
2151:
2149:
2148:
2139:
2135:spinning sidle
2134:
2109:
2105:
2104:
2094:
2089:
2084:
2079:
2074:
2072:
2071:
2062:
2057:
2032:
2028:
2027:
2017:
2012:
2007:
2002:
1997:
1995:
1994:
1985:
1980:
1954:
1949:
1948:
1938:
1933:
1928:
1926:
1925:
1916:
1911:
1886:
1882:
1881:
1871:
1866:
1861:
1856:
1851:
1849:
1848:
1839:
1834:
1808:
1803:
1802:
1792:
1787:
1782:
1780:
1779:
1764:
1730:
1725:
1719:
1717:Euclidean plane
1704:
1686:
1682:
1678:
1674:
1656:
1652:
1648:
1629:
1625:
1606:
1602:
1583:
1579:
1575:
1571:
1552:
1548:
1544:
1525:
1507:
1503:
1499:
1481:
1477:
1458:
1439:
1420:
1416:
1412:
1393:
1389:
1365:
1347:
1343:
1339:
1321:
1317:
1298:
1279:
1260:
1256:
1252:
1233:
1229:
1185:
1166:
1163:
1128:
1093:
1071:Hermann–Mauguin
1056:
1047:
1036:
1027:
1018:
1008:
1002:
996:
987:
978:
969:
960:
951:
942:
932:
926:
920:
914:
908:
902:
898:
889:
880:
871:
862:
853:
844:
834:
828:
822:
816:
810:
804:
800:
789:
784:
724:
723:
722:
721:
720:
713:
705:
704:
697:
689:
688:
681:
670:
645:
610:
607:
606:
583:
572:
570:
568:
565:
564:
532:
530:
528:
525:
524:
506:
486:
442:
408:
405:
404:
385:
382:
381:
303:
300:
299:
257:
254:
253:
211:
208:
207:
172:
150:
146:
144:
141:
140:
119:
115:
113:
110:
109:
106:Euclidean plane
80:
76:
74:
71:
70:
56:Euclidean space
44:symmetry groups
17:
12:
11:
5:
3897:
3887:
3886:
3881:
3876:
3871:
3857:
3856:
3850:
3834:
3833:External links
3831:
3830:
3829:
3783:
3765:John H. Conway
3762:
3751:
3748:
3738:
3737:
3728:
3719:
3707:
3697:
3696:
3694:
3691:
3690:
3689:
3680:
3673:
3670:
3667:
3666:
3662:
3661:
3658:
3655:
3651:
3650:
3647:
3644:
3628:
3627:
3624:
3623:*2 3 105, *257
3621:
3605:
3604:
3601:
3598:
3582:
3581:
3578:
3575:
3545:
3544:
3541:
3538:
3534:
3533:
3530:
3527:
3523:
3522:
3519:
3516:
3512:
3511:
3508:
3505:
3501:
3500:
3498:
3495:
3479:
3478:
3476:
3473:
3469:
3468:
3465:
3462:
3458:
3457:
3454:
3451:
3447:
3446:
3443:
3440:
3436:
3435:
3433:
3430:
3426:
3425:
3422:
3419:
3415:
3414:
3412:
3409:
3405:
3404:
3402:
3399:
3395:
3394:
3392:
3389:
3385:
3384:
3382:
3379:
3375:
3374:
3369:
3366:
3352:
3351:
3340:
3329:
3318:
3309:
3297:
3296:
3285:
3274:
3263:
3251:
3250:
3239:
3228:
3217:
3206:
3194:
3193:
3189:
3188:
3177:
3168:
3159:
3148:
3136:
3135:
3124:
3115:
3106:
3097:
3085:
3084:
3080:
3079:
3070:
3059:
3050:
3041:
3029:
3028:
3017:
3006:
2997:
2986:
2974:
2973:
2962:
2953:
2942:
2930:
2929:
2913:
2910:
2907:
2906:
2903:
2900:
2897:
2894:
2891:
2888:
2885:
2883:
2879:
2878:
2875:
2872:
2868:
2865:
2862:
2859:
2856:
2854:
2850:
2849:
2846:
2843:
2839:
2836:
2833:
2830:
2827:
2825:
2821:
2820:
2817:
2814:
2810:
2807:
2804:
2801:
2798:
2796:
2792:
2791:
2788:
2785:
2782:
2779:
2776:
2773:
2770:
2768:
2764:
2763:
2760:
2757:
2753:
2750:
2747:
2744:
2741:
2739:
2735:
2734:
2731:
2728:
2724:
2721:
2718:
2715:
2712:
2710:
2706:
2705:
2702:
2699:
2695:
2692:
2689:
2686:
2683:
2681:
2677:
2676:
2673:
2670:
2666:
2663:
2660:
2657:
2654:
2652:
2648:
2647:
2644:
2641:
2638:
2635:
2632:
2629:
2626:
2624:
2620:
2619:
2616:
2613:
2610:
2607:
2604:
2601:
2598:
2596:
2592:
2591:
2588:
2585:
2582:
2579:
2576:
2573:
2570:
2568:
2564:
2563:
2560:
2557:
2554:
2551:
2548:
2545:
2542:
2540:
2536:
2535:
2532:
2529:
2526:
2523:
2520:
2517:
2514:
2512:
2508:
2507:
2504:
2501:
2498:
2495:
2492:
2489:
2486:
2484:
2480:
2479:
2476:
2473:
2470:
2467:
2464:
2461:
2458:
2456:
2452:
2451:
2448:
2445:
2442:
2439:
2436:
2433:
2430:
2428:
2424:
2423:
2418:
2413:
2404:
2397:
2392:
2376:
2375:
2368:
2360:
2359:
2356:
2352:
2351:
2344:
2336:
2335:
2332:
2322:
2319:
2318:
2317:
2310:
2304:
2303:
2298:
2297:spinning jump
2289:
2282:
2275:
2272:
2269:
2265:
2260:
2257:
2227:
2223:
2222:
2217:
2208:
2201:
2194:
2191:
2188:
2184:
2179:
2176:
2146:
2142:
2141:
2136:
2127:
2120:
2113:
2110:
2107:
2102:
2099:
2069:
2065:
2064:
2059:
2050:
2043:
2036:
2033:
2030:
2025:
2022:
1992:
1988:
1987:
1982:
1973:
1966:
1959:
1956:
1952:
1946:
1943:
1923:
1919:
1918:
1913:
1904:
1897:
1890:
1887:
1884:
1879:
1876:
1846:
1842:
1841:
1836:
1827:
1820:
1813:
1810:
1806:
1800:
1797:
1777:
1773:
1772:
1769:
1761:
1758:
1753:
1748:
1743:
1729:
1726:
1718:
1715:
1712:
1711:
1708:
1705:
1702:
1699:
1697:
1693:
1692:
1689:
1679:
1676:
1672:
1669:
1667:
1663:
1662:
1659:
1653:
1650:
1646:
1643:
1641:
1637:
1636:
1633:
1630:
1627:
1623:
1620:
1618:
1614:
1613:
1610:
1607:
1604:
1600:
1597:
1595:
1591:
1590:
1587:
1576:
1573:
1569:
1566:
1564:
1560:
1559:
1556:
1549:
1546:
1542:
1539:
1537:
1533:
1532:
1529:
1526:
1523:
1520:
1518:
1514:
1513:
1510:
1500:
1497:
1494:
1492:
1488:
1487:
1484:
1478:
1475:
1472:
1470:
1466:
1465:
1462:
1459:
1456:
1453:
1451:
1447:
1446:
1443:
1440:
1437:
1434:
1432:
1428:
1427:
1424:
1413:
1410:
1407:
1405:
1401:
1400:
1397:
1390:
1387:
1384:
1382:
1378:
1377:
1376:Special cases
1373:
1372:
1369:
1366:
1363:
1360:
1358:
1354:
1353:
1350:
1340:
1337:
1334:
1332:
1328:
1327:
1324:
1318:
1315:
1312:
1310:
1306:
1305:
1302:
1299:
1296:
1293:
1291:
1287:
1286:
1283:
1280:
1277:
1274:
1272:
1268:
1267:
1264:
1253:
1250:
1247:
1245:
1241:
1240:
1237:
1230:
1227:
1224:
1222:
1218:
1217:
1216:= 3, 4, 5 ...
1209:
1208:
1205:
1202:
1199:
1197:
1193:
1192:
1189:
1186:
1183:
1180:
1178:
1174:
1173:
1170:
1164:
1161:
1158:
1156:
1152:
1151:
1148:
1145:
1142:
1140:
1136:
1135:
1132:
1129:
1126:
1123:
1121:
1117:
1116:
1113:
1110:
1107:
1105:
1101:
1100:
1097:
1094:
1091:
1088:
1086:
1082:
1081:
1077:
1076:
1073:
1068:
1063:
1058:
1039:
1038:
1029:
1020:
1010:
1009:
1006:
1003:
1000:
997:
994:
990:
989:
980:
971:
962:
953:
944:
934:
933:
930:
927:
924:
921:
918:
915:
912:
909:
906:
903:
900:
896:
892:
891:
882:
873:
864:
855:
846:
836:
835:
832:
829:
826:
823:
820:
817:
814:
811:
808:
805:
802:
798:
788:
785:
783:
780:
714:
707:
706:
698:
691:
690:
682:
675:
674:
673:
672:
671:
669:
666:
662:
661:
658:
655:
652:
644:
641:
633:
632:
626:
614:
603:
589:
586:
581:
578:
575:
558:
545:
541:
538:
535:
505:
502:
485:
482:
441:
440:Good orbifolds
438:
437:
436:
429:
412:
401:
389:
378:
367:
358:
356:gyration point
328:
327:
307:
296:
281:
272:
261:
244:
233:
230:
227:
224:
221:
218:
215:
192:
191:
188:
185:
182:
179:
171:
168:
153:
149:
122:
118:
83:
79:
15:
9:
6:
4:
3:
2:
3896:
3885:
3882:
3880:
3877:
3875:
3872:
3870:
3867:
3866:
3864:
3854:
3851:
3848:
3844:
3840:
3837:
3836:
3827:
3823:
3819:
3815:
3811:
3807:
3802:
3797:
3793:
3789:
3784:
3782:
3778:
3774:
3770:
3766:
3763:
3760:
3756:
3752:
3749:
3746:
3742:
3741:
3732:
3723:
3716:
3711:
3702:
3698:
3688:
3684:
3681:
3679:
3676:
3675:
3663:
3659:
3656:
3653:
3652:
3648:
3645:
3630:
3629:
3625:
3622:
3607:
3606:
3602:
3599:
3584:
3583:
3579:
3576:
3547:
3546:
3542:
3539:
3536:
3535:
3531:
3529:*2 3 24, *248
3528:
3525:
3524:
3520:
3517:
3514:
3513:
3509:
3506:
3503:
3502:
3499:
3496:
3481:
3480:
3477:
3474:
3471:
3470:
3466:
3463:
3460:
3459:
3455:
3452:
3449:
3448:
3444:
3441:
3438:
3437:
3434:
3431:
3428:
3427:
3423:
3421:*239, *2 3 10
3420:
3417:
3416:
3413:
3410:
3407:
3406:
3403:
3400:
3397:
3396:
3393:
3390:
3387:
3386:
3383:
3380:
3377:
3376:
3373:
3370:
3367:
3364:
3363:
3357:
3350:
3345:
3341:
3339:
3334:
3330:
3328:
3323:
3319:
3314:
3310:
3308:
3303:
3299:
3298:
3295:
3290:
3286:
3284:
3279:
3275:
3273:
3268:
3264:
3262:
3257:
3253:
3252:
3249:
3244:
3240:
3238:
3233:
3229:
3227:
3222:
3218:
3216:
3211:
3207:
3205:
3200:
3196:
3195:
3190:
3187:
3182:
3178:
3173:
3169:
3164:
3160:
3158:
3153:
3149:
3147:
3142:
3138:
3137:
3134:
3129:
3125:
3120:
3116:
3111:
3107:
3102:
3098:
3096:
3091:
3087:
3086:
3081:
3075:
3071:
3069:
3064:
3060:
3055:
3051:
3046:
3042:
3040:
3035:
3031:
3030:
3027:
3022:
3018:
3016:
3011:
3007:
3002:
2998:
2996:
2991:
2987:
2985:
2980:
2976:
2975:
2972:
2967:
2963:
2958:
2954:
2952:
2947:
2943:
2941:
2936:
2932:
2931:
2926:
2923:
2919:
2901:
2895:
2889:
2886:
2884:
2881:
2880:
2873:
2866:
2860:
2857:
2855:
2852:
2851:
2844:
2837:
2831:
2828:
2826:
2823:
2822:
2815:
2808:
2802:
2799:
2797:
2794:
2793:
2786:
2780:
2774:
2771:
2769:
2766:
2765:
2758:
2751:
2745:
2742:
2740:
2737:
2736:
2729:
2722:
2716:
2713:
2711:
2708:
2707:
2700:
2693:
2687:
2684:
2682:
2679:
2678:
2671:
2664:
2658:
2655:
2653:
2650:
2649:
2642:
2636:
2630:
2627:
2625:
2622:
2621:
2614:
2608:
2602:
2599:
2597:
2594:
2593:
2586:
2580:
2574:
2571:
2569:
2566:
2565:
2558:
2552:
2546:
2543:
2541:
2538:
2537:
2530:
2524:
2518:
2515:
2513:
2510:
2509:
2502:
2496:
2490:
2487:
2485:
2482:
2481:
2474:
2468:
2462:
2459:
2457:
2454:
2453:
2446:
2440:
2434:
2431:
2429:
2426:
2425:
2419:
2414:
2412:
2408:
2405:
2403:
2398:
2396:
2393:
2388:
2387:
2384:
2373:
2369:
2366:
2362:
2361:
2357:
2354:
2353:
2349:
2345:
2342:
2338:
2337:
2333:
2330:
2329:
2315:
2311:
2308:
2307:
2299:
2294:
2287:
2283:
2280:
2276:
2273:
2258:
2228:
2225:
2224:
2218:
2213:
2206:
2202:
2199:
2195:
2192:
2177:
2147:
2144:
2143:
2137:
2132:
2125:
2121:
2118:
2114:
2111:
2100:
2070:
2067:
2066:
2060:
2058:spinning hop
2055:
2048:
2044:
2041:
2037:
2034:
2023:
1993:
1990:
1989:
1983:
1978:
1971:
1967:
1964:
1960:
1957:
1955:
1944:
1924:
1921:
1920:
1914:
1909:
1902:
1898:
1895:
1891:
1888:
1877:
1847:
1844:
1843:
1837:
1832:
1825:
1821:
1818:
1814:
1811:
1809:
1798:
1778:
1775:
1774:
1770:
1767:
1763:Examples and
1759:
1757:
1754:
1752:
1749:
1747:
1744:
1742:
1739:
1738:
1735:
1734:Frieze groups
1728:Frieze groups
1724:
1709:
1706:
1700:
1698:
1695:
1694:
1690:
1680:
1670:
1668:
1665:
1664:
1660:
1654:
1644:
1642:
1639:
1638:
1634:
1631:
1621:
1619:
1616:
1615:
1611:
1608:
1598:
1596:
1593:
1592:
1588:
1577:
1567:
1565:
1562:
1561:
1557:
1550:
1540:
1538:
1535:
1534:
1530:
1527:
1521:
1519:
1516:
1515:
1511:
1501:
1495:
1493:
1490:
1489:
1485:
1479:
1473:
1471:
1468:
1467:
1463:
1460:
1454:
1452:
1449:
1448:
1444:
1441:
1435:
1433:
1430:
1429:
1425:
1414:
1408:
1406:
1403:
1402:
1398:
1391:
1385:
1383:
1380:
1379:
1374:
1370:
1367:
1361:
1359:
1356:
1355:
1351:
1341:
1335:
1333:
1330:
1329:
1325:
1319:
1313:
1311:
1308:
1307:
1303:
1300:
1294:
1292:
1289:
1288:
1284:
1281:
1275:
1273:
1270:
1269:
1265:
1254:
1248:
1246:
1243:
1242:
1238:
1231:
1225:
1223:
1220:
1219:
1215:
1210:
1206:
1203:
1200:
1198:
1195:
1194:
1190:
1187:
1181:
1179:
1176:
1175:
1171:
1165:
1159:
1157:
1154:
1153:
1149:
1146:
1143:
1141:
1138:
1137:
1133:
1130:
1124:
1122:
1119:
1118:
1114:
1111:
1108:
1106:
1103:
1102:
1098:
1095:
1089:
1087:
1084:
1083:
1078:
1074:
1072:
1069:
1067:
1064:
1062:
1059:
1054:
1053:
1046:
1034:
1025:
1016:
1011:
991:
985:
981:
976:
972:
967:
963:
958:
954:
949:
945:
940:
936:
935:
928:
922:
916:
910:
904:
894:
893:
887:
883:
878:
874:
869:
865:
860:
856:
851:
847:
842:
838:
837:
830:
824:
818:
812:
806:
796:
795:
779:
777:
772:
770:
765:
762:Similarly, a
760:
758:
754:
750:
746:
742:
738:
733:
729:
718:
711:
702:
695:
686:
679:
665:
659:
656:
653:
650:
649:
648:
640:
636:
630:
627:
612:
605:asterisk and
604:
587:
584:
579:
576:
573:
562:
559:
543:
539:
536:
533:
522:
519:
518:
517:
515:
511:
501:
499:
495:
491:
488:An object is
481:
479:
475:
471:
467:
463:
459:
455:
451:
447:
434:
430:
427:
426:frieze groups
402:
387:
379:
376:
372:
368:
365:
364:
359:
357:
353:
349:
345:
341:
340:
339:
336:
333:
325:
321:
305:
297:
294:
290:
286:
282:
279:
278:
273:
251:
250:
245:
231:
228:
225:
222:
219:
216:
213:
206:
205:
200:
199:
198:
195:
189:
186:
183:
180:
177:
176:
175:
167:
151:
147:
138:
120:
116:
107:
103:
99:
98:frieze groups
81:
77:
68:
64:
59:
57:
53:
49:
45:
41:
37:
33:
29:
27:
22:
3869:Group theory
3791:
3787:
3772:
3758:
3754:
3744:
3731:
3722:
3710:
3701:
3687:space groups
3507:*2 3 18, 239
3355:
3317:*(2∞)
2355:(*333), p3m
2331:(*442), p4m
1771:Description
1755:
1213:
773:
761:
756:
752:
740:
736:
725:
663:
657:*22 and *221
646:
643:Equal groups
637:
634:
631:counts as 2.
628:
560:
520:
507:
493:
489:
487:
477:
473:
469:
465:
461:
457:
453:
449:
445:
443:
432:
374:
370:
361:
351:
343:
337:
329:
319:
292:
288:
284:
275:
247:
202:
196:
193:
173:
63:point groups
60:
31:
24:
18:
3521:, , , , ,
3307:*222∞
2417:Guggenhein
2334:(4*2), p4g
660:2* and 2*1.
403:the symbol
369:an integer
342:an integer
298:the symbol
291:and also a
283:the symbol
40:John Conway
3863:Categories
3801:1910.00519
3693:References
3368:Orbifolds
3365:−1/χ
3133:*33∞
3026:*∞42
2971:*23∞
2420:Fejes Toth
2391:signature
2358:(632), p6
1721:See also:
1551:1/mmm or 2
1392:2/mmm or 2
1232:n/mmm or 2
1066:Schönflies
1057:signature
1043:See also:
1037:Order 120
683:A perfect
654:22 and 221
651:1* and *11
625:count as 1
498:orientable
3847:PDF, 2006
3826:203610179
3580:... , ,
3543:, , , ,
3515:17.5–16.2
3445:, , , ,
2922:triangles
1768:nickname
1055:Orbifold
1028:Order 48
1019:Order 24
1005:(*532), I
999:(*432), O
993:(*332), T
988:Order 24
979:Order 20
970:Order 16
961:Order 12
929:(*226), D
923:(*225), D
917:(*224), D
911:(*223), D
905:(*222), D
895:(*221), D
890:Order 12
881:Order 10
787:Spherical
685:snowflake
613:×
577:−
537:−
472:≥ 2, and
411:∞
388:×
354:around a
350:of order
306:×
260:∞
232:…
201:positive
3672:See also
3649:, , ...
3349:*∞
3327:*∞
3186:*∞
2422:Cadwell
2400:Hermann–
2389:Orbifold
1760:Diagram
1756:Orbifold
1655:1/m or 2
1480:2/m or 2
1320:n/m or 2
952:Order 8
943:Order 4
872:Order 8
863:Order 6
854:Order 4
845:Order 2
831:(*66), C
825:(*55), C
819:(*44), C
813:(*33), C
807:(*22), C
797:(*11), C
728:symmetry
701:pentagon
514:orbifold
363:asterisk
348:rotation
332:boldface
324:crosscap
277:asterisk
252:symbol,
249:infinity
204:integers
52:orbifold
28:notation
26:orbifold
21:geometry
3806:Bibcode
3639:⁄
3616:⁄
3593:⁄
3570:⁄
3556:⁄
3490:⁄
3475:*2 3 16
3467:, , ,
3450:22.3–21
3432:*2 3 11
3418:36–26.4
3372:Coxeter
2407:Speiser
2402:Mauguin
2395:Coxeter
1061:Coxeter
749:digonal
743:•. The
739:• and *
494:achiral
320:miracle
104:of the
96:), the
65:on the
3853:Tegula
3824:
3779:
3775:2008,
2411:Niggli
1981:sidle
1766:Conway
1751:Schön.
1582:2m or
1419:2m or
1259:2m or
1075:Order
745:bullet
512:of an
490:chiral
464:, for
293:handle
289:wonder
67:sphere
3822:S2CID
3796:arXiv
3226:*(24)
3215:*(23)
3204:*2223
3123:*337
3114:*336
3105:*335
3058:*257
3049:*256
3005:*247
2961:*239
2651:*2222
2415:Polya
2274:*22∞
2216:jump
1912:step
755:and *
730:of a
3777:ISBN
3665:...
3497:*247
3472:19.2
3429:26.4
3411:*245
3391:*238
3381:*237
3157:*366
3146:*344
3095:*334
3068:*266
3039:*255
3015:*248
2995:*246
2984:*245
2951:*238
2940:*237
2767:2222
2756:gggg
2727:kkgg
2698:kgkg
2680:2*22
2669:kkkk
2600:p31m
2572:p3m1
2567:*333
2483:*442
2427:*632
2268:×Dih
2226:p2mm
2187:×Dih
2145:p11m
2112:2*∞
2068:p2mg
2035:22∞
1958:*∞∞
1922:p1m1
1845:p11g
1835:hop
1746:Cox.
1381:*222
1221:*22n
1155:*332
1120:*432
1099:120
1085:*532
726:The
715:The
699:The
508:The
446:good
360:the
274:the
246:the
100:and
30:(or
3814:doi
3792:170
3759:165
3626:,
3603:,
3532:,
3510:,
3456:,
3424:,
3401:237
3176:*6
2743:pgg
2738:22×
2714:pmg
2709:22*
2685:cmm
2656:pmm
2623:333
2595:3*3
2539:442
2516:p4g
2511:4*2
2488:p4m
2455:632
2432:p6m
2381:17
2264:Dih
2193:∞*
2106:Dih
2029:Dih
1951:Dih
1889:∞×
1812:∞∞
1741:IUC
1685:or
1675:= C
1649:= C
1626:= C
1603:= C
1572:= C
1545:= C
1536:*22
1506:or
1450:*22
1431:222
1404:2*2
1352:2n
1346:or
1326:2n
1304:2n
1290:*nn
1285:2n
1271:22n
1266:4n
1244:2*n
1239:4n
1207:12
1196:332
1191:24
1177:3*2
1172:24
1150:24
1147:432
1139:432
1134:48
1131:m3m
1115:60
1112:532
1104:532
1096:53m
899:= C
801:= C
759:.)
460:, *
456:, *
380:an
280:, *
166:).
19:In
3865::
3820:,
3812:,
3804:,
3790:,
3771:,
3654:12
3632:12
3618:11
3609:12
3586:13
3563:13
3549:14
3537:15
3526:16
3504:18
3483:18
3461:20
3439:24
3408:40
3398:42
3388:48
3378:84
3338:*2
3294:*2
3283:*2
3272:*2
3261:*2
3248:*4
3237:*3
2887:p1
2871:gg
2858:pg
2853:××
2842:kg
2829:cm
2824:*×
2813:kk
2800:pm
2795:**
2772:p2
2748:2v
2719:2v
2690:2v
2661:2v
2628:p3
2605:3v
2577:3v
2544:p4
2521:4v
2460:p6
2437:6v
2261:∞h
2180:∞h
2103:∞d
1991:p2
1947:∞v
1776:p1
1710:1
1691:2
1666:1×
1661:2
1647:1h
1640:1*
1635:2
1632:1m
1624:1v
1617:*1
1612:2
1609:12
1594:22
1589:4
1574:2h
1570:1d
1563:2*
1558:4
1555:m2
1547:2v
1543:1h
1531:2
1517:22
1512:4
1491:2×
1486:4
1476:2h
1469:2*
1464:4
1461:2m
1457:2v
1445:4
1442:22
1426:8
1411:2d
1399:8
1396:m2
1388:2h
1371:n
1357:nn
1338:2n
1331:n×
1316:nh
1309:n*
1301:nm
1297:nv
1282:n2
1251:nd
1236:m2
1228:nh
1204:23
1188:m3
1169:3m
931:6h
925:5h
919:4h
913:3h
907:2h
901:2v
897:1h
833:6v
827:5v
821:4v
815:3v
809:2v
799:1v
764:1D
757:nn
753:nn
732:2D
480:.
476:≠
468:,
462:pq
454:pq
452:,
23:,
3849:)
3816::
3808::
3798::
3641:7
3637:4
3634:+
3614:8
3611:+
3595:5
3591:1
3588:+
3572:3
3568:1
3565:+
3561:–
3558:5
3554:2
3551:+
3492:3
3488:2
3485:+
2904:1
2902:W
2898:1
2896:C
2892:1
2890:C
2882:o
2876:1
2874:W
2869:1
2867:D
2863:2
2861:C
2847:1
2845:W
2840:1
2838:D
2834:s
2832:C
2818:1
2816:W
2811:1
2809:D
2805:s
2803:C
2789:2
2787:W
2783:2
2781:C
2777:2
2775:C
2761:2
2759:W
2754:2
2752:D
2746:C
2732:2
2730:W
2725:2
2723:D
2717:C
2703:2
2701:W
2696:2
2694:D
2688:C
2674:2
2672:W
2667:2
2665:D
2659:C
2645:3
2643:W
2639:3
2637:C
2633:3
2631:C
2617:3
2615:W
2611:3
2609:D
2603:C
2589:3
2587:W
2583:3
2581:D
2575:C
2561:4
2559:W
2555:4
2553:C
2549:4
2547:C
2533:4
2531:W
2527:4
2525:D
2519:C
2505:4
2503:W
2499:4
2497:D
2493:4
2491:C
2477:6
2475:W
2471:6
2469:C
2465:6
2463:C
2449:6
2447:W
2443:6
2441:D
2435:C
2270:1
2266:∞
2259:D
2189:1
2185:∞
2183:Z
2178:C
2108:∞
2101:D
2031:∞
2026:∞
2024:D
1953:∞
1945:C
1885:∞
1883:Z
1880:∞
1878:S
1807:∞
1805:Z
1801:∞
1799:C
1707:1
1703:1
1701:C
1696:1
1687:1
1683:1
1681:2
1677:i
1673:2
1671:S
1657:1
1651:s
1645:C
1628:s
1622:C
1605:2
1601:1
1599:D
1586:m
1584:1
1580:1
1578:2
1568:D
1553:1
1541:D
1528:2
1524:2
1522:C
1508:2
1504:2
1502:2
1498:4
1496:S
1482:2
1474:C
1455:C
1438:2
1436:D
1423:m
1421:2
1417:2
1415:2
1409:D
1394:2
1386:D
1368:n
1364:n
1362:C
1348:n
1344:n
1342:2
1336:S
1322:n
1314:C
1295:C
1278:n
1276:D
1263:m
1261:n
1257:n
1255:2
1249:D
1234:n
1226:D
1214:n
1201:T
1184:h
1182:T
1167:4
1162:d
1160:T
1144:O
1127:h
1125:O
1109:I
1092:h
1090:I
1007:h
1001:h
995:d
803:s
741:n
737:n
629:o
588:n
585:2
580:1
574:n
561:n
544:n
540:1
534:n
521:n
478:q
474:p
470:q
466:p
458:p
450:p
433:o
375:n
371:n
352:n
344:n
285:o
229:,
226:3
223:,
220:2
217:,
214:1
152:2
148:H
139:(
121:2
117:E
108:(
82:2
78:S
69:(
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