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Baris 1: {{short description\|Wikipedia list article}} {{Titik navigator grup 3D}} ~~'''Kelompok~~Grup simetri bola hingga~~''' disebut~~ juga ~~kelompok~~disebut [[grup titik dalam tiga dimensi]]. ~~Terdapat~~Ada lima kelas simetri ~~fundamental~~dasar yang memiliki domain ~~fundamental~~dasar segitiga: [[~~simetri~~Simetri 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> ~~{{matematika-stub}}~~ [[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> ~~[[Kategori:Geometri]]~~ == 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>). {\| class="wikitable" \|- ! [[Notasi Hermann–Mauguin\|Intl]] ! [[Notasi aljabar geometri\|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 orbifold\|Orb.]] ! [[Notasi Schönflies\|Schön.]] ! [[John Horton Conway\|Con.]] ! [[Notasi Coxeter\|Cox.]] ! Uru. ! [[Domain fundamental\|Dom.<br>Fundamental]] \|- align=center \| 1 \| {{overline\|1}} \| 11 \| C<SUB>1</SUB> \| C<SUB>1</SUB> \| ][<br>[ ]<sup>+</sup> \| 1 \| [[Berkas:Sphere symmetry group c1.png\|100px]] \|- align=center \| 2 \| {{overline\|2}} \| 22 \| D<SUB>1</SUB><br>= C<SUB>2</SUB> \| D<SUB>2</SUB><br>= C<SUB>2</SUB> \| [2]<sup>+</sup> \| 2 \| [[Berkas:Sphere symmetry group c2.png\|100px]] \|- align=center \| {{overline\|1}} \| {{overline\|22}} \| × \| C<SUB>i</SUB><br>= S<SUB>2</SUB> \| CC<SUB>2</SUB> \| [2<sup>+</sup>,2<sup>+</sup>] \| 2 \| [[Berkas:Sphere symmetry group ci.png\|100px]] \|- align=center \| {{overline\|2}}<br>= m \| 1 \| * \| C<SUB>s</SUB><br>= C<SUB>1v</SUB><br>= C<SUB>1h</SUB> \| ±C<SUB>1</SUB><br>= CD<SUB>2</SUB> \| [ ] \| 2 \| [[Berkas: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}} \| 22 \| 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> \| 23<br>24<br>25<br>26<br>2n \| 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}} \| 32 \| 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 == * [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:Polihedra]] [[Kategori:Simetri]] [[Kategori:Teori grup]] [[Kategori:Daftar terkait matematika\|Grup simetri bola]]