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Baris 9:
== Simetri involusional ==
 
Ada empat grup [[Involusi (matematika)|involusiinvolusial]]al: tidak ada simetri (C<sub>1</sub>), [[simetri refleksi]] (C<sub>s</sub>), Simetri rotasi lipatan 2 (C<sub>2</sub>), dan [[simetri titik]] pusat (C<sub>i</sub>).
 
{| class="wikitable"
Baris 29:
| ][<br>[&nbsp;]<sup>+</sup>
| 1
| [[GambarBerkas:Sphere symmetry group c1.png|100px]]
|- align=center
| 2
Baris 38:
| [2]<sup>+</sup>
| 2
| [[GambarBerkas:Sphere symmetry group c2.png|100px]]
|- align=center
| {{overline|1}}
Baris 47:
| [2<sup>+</sup>,2<sup>+</sup>]
| 2
| [[GambarBerkas:Sphere symmetry group ci.png|100px]]
|- align=center
| {{overline|2}}<br>= m
Baris 56:
| [&nbsp;]
| 2
| [[GambarBerkas:Sphere symmetry group cs.png|100px]]
|}
 
== Simetri siklik ==
 
Ada empat famili [[simetri siklik]] tak hingga, dengan ''n''  =  2 atau lebih tinggi. (''n'' mungkin 1 sebagai kasus khusus sebagai ''tidak simetri'')
 
{| class="wikitable"
Baris 81:
| [2<sup>+</sup>,4<sup>+</sup>]
| 4
| [[GambarBerkas:Sphere symmetry group s4.png|100px]]
|- align=center
| 2/m
Baris 90:
| [2,2<sup>+</sup>]<br>[2<sup>+</sup>,2]
| 4
| [[GambarBerkas:Sphere symmetry group c2h.png|100px]]
|}
 
Baris 111:
| [2]<sup>+</sup><br>[3]<sup>+</sup><br>[4]<sup>+</sup><br>[5]<sup>+</sup><br>[6]<sup>+</sup><br>[n]<sup>+</sup><br>
| 2<br>3<br>4<br>5<br>6<br>n
| [[GambarBerkas:Sphere symmetry group c2.png|100px]]
|- align=center valign=top
| 2mm<br>3m<br>4mm<br>5m<br>6mm<br>nm (n adalah nilai ganjil)<br>nmm (n adalah nilai ganda)
Baris 120:
| [2]<br>[3]<br>[4]<br>[5]<br>[6]<br>[n]
| 4<br>6<br>8<br>10<br>12<br>2n
| [[GambarBerkas:Sphere symmetry group c2v.png|100px]]
|- align=center valign=top
| {{overline|3}}<br>{{overline|8}}<br>{{overline|5}}<br>{{overline|12}}<br>-
Baris 129:
| [2<sup>+</sup>,6<sup>+</sup>]<br>[2<sup>+</sup>,8<sup>+</sup>]<br>[2<sup>+</sup>,10<sup>+</sup>]<br>[2<sup>+</sup>,12<sup>+</sup>]<br>[2<sup>+</sup>,2n<sup>+</sup>]
| 6<br>8<br>10<br>12<br>2n
|[[GambarBerkas:Sphere symmetry group s6.png|100px]]
|- align=center valign=top
| 3/m={{overline|6}}<br>4/m<br>5/m={{overline|10}}<br>6/m<br>n/m
Baris 138:
| [2,3<sup>+</sup>]<br>[2,4<sup>+</sup>]<br>[2,5<sup>+</sup>]<br>[2,6<sup>+</sup>]<br>[2,n<sup>+</sup>]
| 6<br>8<br>10<br>12<br>2n
|[[GambarBerkas:Sphere symmetry group c3h.png|100px]]
|}
 
Baris 163:
| [2,2]<sup>+</sup>
| 4
|[[GambarBerkas:Sphere symmetry group d2.png|100px]]
|- align=center
| {{Overline|4}}2m
Baris 172:
| [2<sup>+</sup>,4]
| 8
| [[GambarBerkas:Sphere symmetry group d2d.png|100px]]
|- align=center
| mmm
Baris 181:
| [2,2]
| 8
| [[GambarBerkas:Sphere symmetry group d2h.png|100px]]
|}
 
Baris 202:
| [2,3]<sup>+</sup><br>[2,4]<sup>+</sup><br>[2,5]<sup>+</sup><br>[2,6]<sup>+</sup><br>[2,n]<sup>+</sup>
| 6<br>8<br>10<br>12<br>2n
|[[GambarBerkas:Sphere symmetry group d3.png|100px]]
|- align=center valign=top
| {{Overline|3}}m<br>{{Overline|8}}2m<br>{{Overline|5}}m<br>{{Overline|12}}.2m<br>
Baris 211:
| [2<sup>+</sup>,6]<br>[2<sup>+</sup>,8]<br>[2<sup>+</sup>,10]<br>[2<sup>+</sup>,12]<br>[2<sup>+</sup>,2n]
| 12<br>16<br>20<br>24<br>4n
| [[GambarBerkas:Sphere symmetry group d3d.png|100px]]
|- align=center valign=top
| {{overline|6}}m2<br>4/mmm<br>{{overline|10}}m2<br>6/mmm
Baris 220:
| [2,3]<br>[2,4]<br>[2,5]<br>[2,6]<br>[2,n]
| 12<br>16<br>20<br>24<br>4n
| [[GambarBerkas:Sphere symmetry group d3h.png|100px]]
|}
 
Baris 246:
| [3,3]<sup>+</sup><br>= [4,3<sup>+</sup>]<sup>+</sup>
| 12
| [[GambarBerkas:Sphere symmetry group t.png|100px]]
|- align=center
| m{{overline|3}}
Baris 255:
| [4,3<sup>+</sup>]
| 24
| [[GambarBerkas:Sphere symmetry group th.png|100px]]
|- align=center
| {{overline|4}}3m
Baris 264:
| [3,3]<br>= [1<sup>+</sup>,4,3]
| 24
| [[GambarBerkas:Sphere symmetry group td.png|100px]]
|}
 
Baris 286:
| [4,3]<sup>+</sup><br>= <nowiki>[[3</nowiki>,3]]<sup>+</sup>
| 24
| [[GambarBerkas:Sphere symmetry group o.png|100px]]
|- align=center
| m{{overline|3}}m
Baris 295:
| [4,3]<br>= <nowiki>[[</nowiki>3,3]]
| 48
| [[GambarBerkas:Sphere symmetry group oh.png|100px]]
|}
 
Baris 317:
| [5,3]<sup>+</sup>
| 60
| [[GambarBerkas:Sphere symmetry group i.png|100px]]
|- align=center
| {{overline|53}}2/m
Baris 326:
| [5,3]
| 120
| [[GambarBerkas:Sphere symmetry group ih.png|100px]]
|}
 
== Lihat pula ==
* [[Grup titik kristalografi]]
* [[Grup segitiga]]
* [[Daftar grup simetri planar]]
* [[Grup titik dalam dua dimensi]]
 
== Catatan ==
{{reflist}}
 
== Referensi ==
* Peter R. Cromwell, ''Polyhedra'' (1997), Appendix I
* {{cite book|last=Sands |first=Donald E. |title=Introduction to Crystallography |url=https://archive.org/details/introductiontocr0000sand |year=1993 |publisher=Dover Publications, Inc. |location=Mineola, New York |isbn=0-486-67839-3 |chapter=Crystal Systems and Geometry |page=[https://archive.org/details/introductiontocr0000sand/page/165 165] }}
* ''On Quaternions and Octonions'', 2003, John Horton Conway and Derek A. Smith {{isbn|978-1-56881-134-5}}
* ''The Symmetries of Things'' 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, {{isbn|978-1-56881-220-5}}
* '''Kaleidoscopes: Selected Writings of [[Harold Scott MacDonald Coxeter|H.S.M. Coxeter]]''', edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, {{isbn|978-0-471-01003-6}} [http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html]
Baris 349:
* [[Norman Johnson (mathematician)|N.W. Johnson]]: ''Geometries and Transformations'', (2018) {{ISBN|978-1-107-10340-5}} Chapter 11: ''Finite symmetry groups'', Table 11.4 Finite Groups of Isometries in 3-space
 
== Pranala luar ==
* [http://www.geom.uiuc.edu/~math5337/Orbifolds/costs.html Finite spherical symmetry groups]
* {{MathWorld | urlname=SchoenfliesSymbol | title=Schoenflies symbol}}
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