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{{short description|Wikipedia list article}}
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'''KelompokGrup simetri bola hingga''' disebut juga kelompokdisebut [[grup titik dalam tiga dimensi]]. TerdapatAda lima kelas simetri fundamentaldasar yang memiliki domain fundamentaldasar segitiga: [[simetriSimetri dihedral|dihedral]], [[simetri siklik|siklik]], [[simetri tetrahedral|tetrahedral]], [[simetri oktahedral|oktahedral]], dan simetri [[simetri ikosahedral|ikosahedral]].
 
Artikel ini mencantumkan grup menurut [[notasi Schoenflies]], [[notasi Coxeter]],<ref>Johnson, 2015</ref> [[notasi orbifold]],<ref>Conway, 2008</ref> dan urutan. [[John Horton Conway|John Conway]] menggunakan variasi dari notasi Schoenflies, berdasarkan struktur aljabar grup [[kuaternion]], diberikan label oleh satu atau dua huruf besar, dan subskrip bilangan bulat. Urutan grup didefinisikan sebagai subskrip, kecuali urutannya digandakan untuk simbol dengan plus atau minus, "±", awalan yang menyiratkan [[inversi pusat]].<ref>Conway, 2003</ref>
 
[[Notasi Hermann–Mauguin]] (notasi internasional) juga diberikan. Grup [[kristalografi]], total 32, adalah himpunan bagian dengan urutan elemen 2, 3, 4 dan 6.<ref>Sands, 1993</ref>
 
== Simetri involusional ==
 
Ada empat grup [[Involusi (matematika)|involusial]]: 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>).
 
== Simetri konstitusional ==
{| class="wikitable"
|-
! [[Notasi Hermann–Mauguin|Intl]]
! [[Notasi Geometrialjabar Aljabargeometri|Geo]]<br><ref>''The Crystallographic Space groups in Geometric algebra'', [[D. Hestenes]] and J. Holt, Journal of Mathematical Physics. 48, 023514 (2007) (22 pages) [[PDF]] [http://geocalc.clas.asu.edu/pdf/CrystalGA.pdf]</ref>
! [[Notasi Orbifoldorbifold|Orb.]]
! [[Notasi Schönflies|Schön.]]
! [[John Horton Conway|Con.]]
! [[Notasi Coxeter|Cox.]]
! OrdUru.
! [[Domain dasarfundamental|Dom.<br>dasarFundamental]]
|- align=center
| 1
Baris 21 ⟶ 29:
| ][<br>[&nbsp;]<sup>+</sup>
| 1
| [[GambarBerkas:Sphere symmetry group c1.png|100px]]
|- align=center
| 2
Baris 30 ⟶ 38:
| [2]<sup>+</sup>
| 2
| [[GambarBerkas:Sphere symmetry group c2.png|100px]]
|- align=center
| {{overline|1}}
| {{overline|22}}
| &times;×
| C<SUB>i</SUB><br>= S<SUB>2</SUB>
| CC<SUB>2</SUB>
| [2<sup>+</sup>,2<sup>+</sup>]
| 2
| [[GambarBerkas:Sphere symmetry group ci.png|100px]]
|- align=center
| {{overline|2}}<br>= m
Baris 48 ⟶ 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"
|-
! [[Notasi Hermann–Mauguin|Intl]]
! Geo<br>
! [[Notasi orbifold|Orb.]]
! [[Notasi Schönflies|Schön.]]
! [[John Horton Conway|Con.]]
! [[Notasi Coxeter|Cox.]]
! Uru.
! [[Domain fundamental|Dom.<br>fundamental]]
|- align=center
| {{overline|4}}
| {{overline|42}}
| 2×
| S<SUB>4</SUB>
| CC<SUB>4</SUB>
| [2<sup>+</sup>,4<sup>+</sup>]
| 4
| [[Berkas:Sphere symmetry group s4.png|100px]]
|- align=center
| 2/m
| {{overline|2}}2
| 2*
| C<SUB>2h</SUB><br>= D<SUB>1d</SUB>
| ±C<SUB>2</SUB><br>= ±D<SUB>2</SUB>
| [2,2<sup>+</sup>]<br>[2<sup>+</sup>,2]
| 4
| [[Berkas:Sphere symmetry group c2h.png|100px]]
|}
 
{| class="wikitable"
|-
! [[Notasi Hermann–Mauguin|Intl]]
! Geo<br>
! [[Notasi orbifold|Orb.]]
! [[Notasi Schönflies|Schön.]]
! [[John Horton Conway|Con.]]
! [[Notasi Coxeter|Cox.]]
! Ord.
! [[Domain fundamental|Dom.<br>fundamental]]
|- align=center valign=top
| 2<br>3<br>4<br>5<br>6<br>n
| {{overline|2}}<br>{{overline|3}}<br>{{overline|4}}<br>{{overline|5}}<br>{{overline|6}}<br>{{overline|n}}
| 22<br>33<br>44<br>55<br>66<br>nn
| C<sub>2</sub><br>C<sub>3</sub><br>C<sub>4</sub><br>C<sub>5</sub><br>C<sub>6</sub><br>C<sub>n</sub>
| C<sub>2</sub><br>C<sub>3</sub><br>C<sub>4</sub><br>C<sub>5</sub><br>C<sub>6</sub><br>C<sub>n</sub>
| [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
| [[Berkas: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)
| 2<br>3<br>4<br>5<br>6<br>n
| *22<br>*33<br>*44<br>*55<br>*66<br>*nn
| C<SUB>2v</SUB><br>C<SUB>3v</SUB><br>C<SUB>4v</SUB><br>C<SUB>5v</SUB><br>C<SUB>6v</SUB><br>C<SUB>nv</SUB>
| CD<SUB>4</SUB><br>CD<SUB>6</SUB><br>CD<SUB>8</SUB><br>CD<SUB>10</SUB><br>CD<SUB>12</SUB><br>CD<SUB>2n</SUB>
| [2]<br>[3]<br>[4]<br>[5]<br>[6]<br>[n]
| 4<br>6<br>8<br>10<br>12<br>2n
| [[Berkas:Sphere symmetry group c2v.png|100px]]
|- align=center valign=top
| {{overline|3}}<br>{{overline|8}}<br>{{overline|5}}<br>{{overline|12}}<br>-
| {{overline|62}}<br>{{overline|82}}<br>{{overline|10.2}}<br>{{overline|12.2}}<br>{{overline|2n.2}}
| 3×<br>4×<br>5×<br>6×<br>n×
| S<SUB>6</SUB><br>S<SUB>8</SUB><br>S<SUB>10</SUB><br>S<SUB>12</SUB><br>S<SUB>2n</SUB>
| ±C<SUB>3</SUB><br>CC<SUB>8</SUB><br>±C<SUB>5</SUB><br>CC<SUB>12</SUB><br>CC<SUB>2n</SUB> / ±C<SUB>n</SUB>
| [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
|[[Berkas: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
| {{overline|3}}2<br>{{overline|4}}2<br>{{overline|5}}2<br>{{overline|6}}2<br>{{overline|n}}2
| 3*<br>4*<br>5*<br>6*<br>n*
| C<SUB>3h</SUB><br>C<SUB>4h</SUB><br>C<SUB>5h</SUB><br>C<SUB>6h</SUB><br>C<SUB>nh</SUB>
| CC<SUB>6</SUB><br>±C<SUB>4</SUB><br>CC<SUB>10</SUB><br>±C<SUB>6</SUB><br>±C<SUB>n</SUB> / CC<SUB>2n</SUB>
| [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
|[[Berkas:Sphere symmetry group c3h.png|100px]]
|}
 
== Simetri dihedral ==
 
Ada tiga famili [[simetri dihedral dalam tiga dimensi|simetri dihedral]] tak hingga, dengan ''n'' = 2 atau tinggi (''n'' mungkin 1 sebagai kasus khusus).
 
{| class="wikitable"
|-
! [[Notasi Hermann–Mauguin|Intl]]
! Geo<br>
! [[Notasi orbifold|Orb.]]
! [[Notasi Schönflies|Schön.]]
! [[John Horton Conway|Con.]]
! [[Notasi Coxeter|Cox.]]
! Uru.
! [[Domain fundamental|Dom.<br>fundamental]]
|- align=center
| 222
| {{overline|2}}.{{overline|2}}
| 222
| D<SUB>2</SUB>
| D<SUB>4</SUB>
| [2,2]<sup>+</sup>
| 4
|[[Berkas:Sphere symmetry group d2.png|100px]]
|- align=center
| {{Overline|4}}2m
| 4{{overline|2}}
| 2*2
| D<SUB>2d</SUB>
| DD<SUB>8</SUB>
| [2<sup>+</sup>,4]
| 8
| [[Berkas:Sphere symmetry group d2d.png|100px]]
|- align=center
| mmm
| 22
| *222
| D<SUB>2h</SUB>
| ±D<SUB>4</SUB>
| [2,2]
| 8
| [[Berkas:Sphere symmetry group d2h.png|100px]]
|}
 
{| class="wikitable"
|-
! [[Notasi Hermann–Mauguin|Intl]]
! Geo<br>
! [[Notasi orbifold|Orb.]]
! [[Notasi Schönflies|Schön.]]
! [[John Horton Conway|Con.]]
! [[Notasi Coxeter|Cox.]]
! Uru.
! [[Domain fundamental|Dom.<br>fundamental]]
|- align=center valign=top
| 32<br>422<br>52<br>622
| {{overline|3}}.{{overline|2}}<br>{{overline|4}}.{{overline|2}}<br>{{overline|5}}.{{overline|2}}<br>{{overline|6}}.{{overline|2}}<br>{{overline|n}}.{{overline|2}}
| 223<br>224<br>225<br>226<br>22n
| D<SUB>3</SUB><br>D<SUB>4</SUB><br>D<SUB>5</SUB><br>D<SUB>6</SUB><br>D<SUB>n</SUB>
| D<SUB>6</SUB><br>D<SUB>8</SUB><br>D<SUB>10</SUB><br>D<SUB>12</SUB><br>D<SUB>2n</SUB>
| [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
|[[Berkas: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>
| 6{{overline|2}}<br>8{{overline|2}}<br>10.{{overline|2}}<br>12.{{overline|2}}<br>n{{overline|2}}<br>
| 2*3<br>2*4<br>2*5<br>2*6<br>2*n
| D<SUB>3d</SUB><br>D<SUB>4d</SUB><br>D<SUB>5d</SUB><br>D<SUB>6d</SUB><br>D<SUB>nd</SUB>
| ±D<SUB>6</SUB><br>DD<SUB>16</SUB><br>±D<SUB>10</SUB><br>DD<SUB>24</SUB><br>DD<SUB>4n</SUB> / ±D<SUB>2n</SUB>
| [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
| [[Berkas:Sphere symmetry group d3d.png|100px]]
|- align=center valign=top
| {{overline|6}}m2<br>4/mmm<br>{{overline|10}}m2<br>6/mmm
| 32<br>42<br>52<br>62<br>n2
| *223<br>*224<br>*225<br>*226<br>*22n
| D<SUB>3h</SUB><br>D<SUB>4h</SUB><br>D<SUB>5h</SUB><br>D<SUB>6h</SUB><br>D<SUB>nh</SUB>
| DD<SUB>12</SUB><br>±D<SUB>8</SUB><br>DD<SUB>20</SUB><br>±D<SUB>12</SUB><br>±D<SUB>2n</SUB> / DD<SUB>4n</SUB>
| [2,3]<br>[2,4]<br>[2,5]<br>[2,6]<br>[2,n]
| 12<br>16<br>20<br>24<br>4n
| [[Berkas:Sphere symmetry group d3h.png|100px]]
|}
 
== Simetri polihedral ==
{{See|Grup polihedral}}
Ada tiga jenis [[simetri polihedral]]: [[simetri tetrahedral]], [[simetri oktahedral]], dan [[simetri ikosahedral]], dinamai berdasarkan segitiga wajah [[polyhedra biasa]] dengan simetri ini.
 
{| class="wikitable"
|+ [[Simetri tetrahedral]]
|-
! [[Notasi Hermann–Mauguin|Intl]]
! Geo<br>
! [[Notasi orbifold|Orb.]]
! [[Notasi Schönflies|Schön.]]
! [[John Horton Conway|Con.]]
! [[Notasi Coxeter|Cox.]]
! Uru.
! [[Domain fundamental|Dom.<br>fundamental]]
|- align=center
| 23
| {{overline|3}}.{{overline|3}}
| 332
| T
| T
| [3,3]<sup>+</sup><br>= [4,3<sup>+</sup>]<sup>+</sup>
| 12
| [[Berkas:Sphere symmetry group t.png|100px]]
|- align=center
| m{{overline|3}}
| 4{{overline|3}}
| 3*2
| T<SUB>h</SUB>
| ±T
| [4,3<sup>+</sup>]
| 24
| [[Berkas:Sphere symmetry group th.png|100px]]
|- align=center
| {{overline|4}}3m
| 33
| *332
| T<SUB>d</SUB>
| TO
| [3,3]<br>= [1<sup>+</sup>,4,3]
| 24
| [[Berkas:Sphere symmetry group td.png|100px]]
|}
 
{| class="wikitable"
|+ [[Simetri oktahedral]]
|-
! Intl
! Geo
! Orb.
! Schön.
! Con.
! Cox.
! Uru.
! Dom.<br>fundamental
|- align=center
| 432
| {{overline|4}}.{{overline|3}}
| 432
| O
| O
| [4,3]<sup>+</sup><br>= <nowiki>[[3</nowiki>,3]]<sup>+</sup>
| 24
| [[Berkas:Sphere symmetry group o.png|100px]]
|- align=center
| m{{overline|3}}m
| 43
| *432
| O<SUB>h</SUB>
| ±O
| [4,3]<br>= <nowiki>[[</nowiki>3,3]]
| 48
| [[Berkas:Sphere symmetry group oh.png|100px]]
|}
 
{| class="wikitable"
|+ [[Simetri ikosahedral]]
|-
! Intl
! Geo
! Orb.
! Schön.
! Con.
! Cox.
! Uru.
! Dom.<br>fundamental
|- align=center
| 532
| {{overline|5}}.{{overline|3}}
| 532
| I
| I
| [5,3]<sup>+</sup>
| 60
| [[Berkas:Sphere symmetry group i.png|100px]]
|- align=center
| {{overline|53}}2/m
| 53
| *532
| I<SUB>h</SUB>
| ±I
| [5,3]
| 120
| [[Berkas: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]
** (Paper 22) H.S.M. Coxeter, ''Regular and Semi Regular Polytopes I'', [Math. Zeit. 46 (1940) 380–407, MR 2,10]
** (Paper 23) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes II'', [Math. Zeit. 188 (1985) 559–591]
** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'', [Math. Zeit. 200 (1988) 3–45]
* [[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 ==
{{matematika-stub}}
* [http://www.geom.uiuc.edu/~math5337/Orbifolds/costs.html Finite spherical symmetry groups]
* {{MathWorld | urlname=SchoenfliesSymbol | title=Schoenflies symbol}}
* {{MathWorld | urlname=CrystallographicPointGroups | title=Crystallographic point groups}}
* [https://web.archive.org/web/20080316083237/http://homepage.mac.com/dmccooey/polyhedra/Simplest.html Simplest Canonical Polyhedra of Each Symmetry Type], by David I. McCooey
 
[[Kategori:GeometriPolihedra]]
[[Kategori:Simetri]]
[[Kategori:Teori grup]]
[[Kategori:Daftar terkait matematika|Grup simetri bola]]
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