Grup titik: Perbedaan antara revisi

|[[Gambar:Flag of Hong Kong.svg|240px]]<BRbr>''[[Bauhinia blakeana]]'' bunga di bendera wilayah [[Hong Kong]] memiliki simetri C₅; bintang di setiap kelopak memiliki simetri D₅.

|[[Berkas:Yin and Yang.svg|160px]]<BRbr>[[Yin dan Yang]] simbol memiliki geometri simetri C₂ dengan warna terbalik

Dalam [[geometri]], '''grup titik''' adalah [[grup (matematika) | grup]] geometris [[simetri | simetri]] ([[isometri | isometri]]) yang menjaga setidaknya satu titik tetap. Kelompok titik dapat ada dalam [[ruang Euklides]] dengan dimensi apa pun, dan setiap kelompok titik dalam dimensi '' d '' adalah subkelompok dari [[grup ortogonal]] O(''d''). Kelompok titik dapat direalisasikan sebagai himpunan [[matriks ortogonal | matriks ortogonal]] '' M '' yang mengubah titik '' x '' menjadi titik '' y '':

dimana asal adalah titik tetap. Elemen kelompok titik dapat berupa [[Rotasi (matematika) | rotasi]] ([[determinan]] dari ''M'' = 1) atau yang lain [[Refleksi (matematika) | refleksi]], atau [[rotasi tidak tepat]] (determinan dari '' M '' = −1).

Kelompok titik diskrit di lebih dari satu dimensi datang dalam keluarga tak berhingga, tetapi dari [[teorema pembatasan kristalografi]] dan [[Grup ruang#Teorema Bieberbach.27s | salah satu teorema Bieberbach]]. setiap jumlah dimensi hanya memiliki jumlah terbatas dari kelompok titik yang simetris di beberapa [[kisi (grup) | kisi]] ​​atauatau kisi dengan nomor itu. Ini adalah [[grup titik kristalografi]].

! [[Diagram Coxeter–Dynkin|Diagram Coxeter]]

||''C''₁||[&nbsp;]⁺|| ||1||IdentitasIdentity

||''D''₁||[&nbsp;]||{{CDD|node|}}||2||grupReflection refleksigroup

! [[Notasi Hermann–Mauguin | Intl]]

! [[Notasi Coxeter | Coxeter]]

! colspan=2 | [[Diagram Coxeter–Dynkin | Diagram Coxeter]]

!C_1v<BRbr>Urutan 2

!C_2v<BRbr>Urutan 4

!C_3v<BRbr>Urutan 6

!C_4v<BRbr>Urutan 8

!C_5v<BRbr>Urutan 10

!C_6v<BRbr>Urutan 12

!D_1h<BRbr>Urutan 4

!D_2h<BRbr>Urutan 8

!D_3h<BRbr>Urutan 12

!D_4h<BRbr>Urutan 16

!D_5h<BRbr>Urutan 20

!D_6h<BRbr>Urutan 24

!T_d<BRbr>Urutan 24

!O_h<BRbr>Urutan 48

!I_h<BRbr>Urutan 120

! Geo<BRbr><ref>''Grup Ruang Kristalografi dalam aljabar geometris'', [[David Hestenes|D. Hestenes]] and J. Holt, Journal of Mathematical Physics. 48, 023514 (2007) (22 pages) [[PDF]] [http://geocalc.clas.asu.edu/pdf/CrystalGA.pdf] {{Webarchive|url=https://web.archive.org/web/20201020021821/http://geocalc.clas.asu.edu/pdf/CrystalGA.pdf |date=2020-10-20 }}</ref>

! [[Notasi Orbifold | Orbifold]]

! [[Notasi Schönflies | Schönflies]]

! [[John Horton Conway | Conway]]

! [[Notasi Coxeter | Coxeter]]

| 2<BRbr>3<BRbr>4<BRbr>5<BRbr>6<BRbr>n

| {{overline|2}}<BRbr>{{overline|3}}<BRbr>{{overline|4}}<BRbr>{{overline|5}}<BRbr>{{overline|6}}<BRbr>{{overline|n}}

| 22<BRbr>33<BRbr>44<BRbr>55<BRbr>66<BRbr>nn

| C<SUB>2</SUB><BRbr>C<SUB>3</SUB><BRbr>C<SUB>4</SUB><BRbr>C<SUB>5</SUB><BRbr>C<SUB>6</SUB><BRbr>C<SUB>n</SUB>

| C<SUB>2</SUB><BRbr>C<SUB>3</SUB><BRbr>C<SUB>4</SUB><BRbr>C<SUB>5</SUB><BRbr>C<SUB>6</SUB><BRbr>C<SUB>n</SUB>

| [2]⁺<BRbr>[3]⁺<BRbr>[4]⁺<BRbr>[5]⁺<BRbr>[6]⁺<BRbr>[n]⁺<BRbr>

| 2<BRbr>3<BRbr>4<BRbr>5<BRbr>6<BRbr>''n''

| mm2<BRbr>3m<BRbr>4mm<BRbr>5m<BRbr>6mm<br>nmm<br>nm

| 2<BRbr>3<BRbr>4<BRbr>5<BRbr>6<BRbr>n

| *22<BRbr>*33<BRbr>*44<BRbr>*55<BRbr>*66<BRbr>*nn

| C<SUB>2v</SUB><BRbr>C<SUB>3v</SUB><BRbr>C<SUB>4v</SUB><BRbr>C<SUB>5v</SUB><BRbr>C<SUB>6v</SUB><BRbr>C<SUB>nv</SUB>

| CD<SUB>4</SUB><BRbr>CD<SUB>6</SUB><BRbr>CD<SUB>8</SUB><BRbr>CD<SUB>10</SUB><BRbr>CD<SUB>12</SUB><BRbr>CD<SUB>2n</SUB>

| [2]<BRbr>[3]<BRbr>[4]<BRbr>[5]<BRbr>[6]<BRbr>[n]

| 4<BRbr>6<BRbr>8<BRbr>10<BRbr>12<BRbr>2''n''

| 2/m<BRbr>{{overline|6}}<BRbr>4/m<BRbr>{{overline|10}}<BRbr>6/m<BRbr>n/m<BRbr>{{overline|2n}}

| {{overline|2}} 2<BRbr>{{overline|3}} 2<BRbr>{{overline|4}} 2<BRbr>{{overline|5}} 2<BRbr>{{overline|6}} 2<BRbr>{{overline|n}} 2

| 2*<BRbr>3*<BRbr>4*<BRbr>5*<BRbr>6*<BRbr>n*

| C<SUB>2h</SUB><BRbr>C<SUB>3h</SUB><BRbr>C<SUB>4h</SUB><BRbr>C<SUB>5h</SUB><BRbr>C<SUB>6h</SUB><BRbr>C<SUB>nh</SUB>

| ±C<SUB>2</SUB><BRbr>CC<SUB>6</SUB><BRbr>±C<SUB>4</SUB><BRbr>CC<SUB>10</SUB><BRbr>±C<SUB>6</SUB><BRbr>±C<SUB>n</SUB> / CC<SUB>2n</SUB>

| [2,2⁺]<BRbr>[2,3⁺]<BRbr>[2,4⁺]<BRbr>[2,5⁺]<BRbr>[2,6⁺]<BRbr>[2,n⁺]

| 4<BRbr>6<BRbr>8<BRbr>10<BRbr>12<BRbr>2''n''

| {{overline|4}}<BRbr>{{overline|3}}<BRbr>{{overline|8}}<BRbr>{{overline|5}}<BRbr>{{overline|12}}<br>{{overline|2n}}<br>{{overline|n}}

| {{overline|4 2}}<BRbr>{{overline|6 2}}<BRbr>{{overline|8 2}}<BRbr>{{overline|10 2}}<BRbr>{{overline|12 2}}<BRbr>{{overline|2n 2}}

| S<SUB>4</SUB><BRbr>S<SUB>6</SUB><BRbr>S<SUB>8</SUB><BRbr>S<SUB>10</SUB><BRbr>S<SUB>12</SUB><BRbr>S<SUB>2n</SUB>

| CC<SUB>4</SUB><BRbr>±C<SUB>3</SUB><BRbr>CC<SUB>8</SUB><BRbr>±C<SUB>5</SUB><BRbr>CC<SUB>12</SUB><BRbr>CC<SUB>2n</SUB> / ±C<SUB>n</SUB>

| [2⁺,4⁺]<BRbr>[2⁺,6⁺]<BRbr>[2⁺,8⁺]<BRbr>[2⁺,10⁺]<BRbr>[2⁺,12⁺]<BRbr>[2⁺,2n⁺]

| 4<BRbr>6<BRbr>8<BRbr>10<BRbr>12<BRbr>2''n''

! [[Notasi Herman–Maugin| Intl]]

! [[Notasi Schönflies | Schönflies]]

! [[John Horton Conway | Conway]]

! [[Notasi Coxeter | Coxeter]]

| 222<BRbr>32<BRbr>422<BRbr>52<BRbr>622<br>n22<br>n2

| {{overline|2}} {{overline|2}}<BRbr>{{overline|3}} {{overline|2}}<BRbr>{{overline|4}} {{overline|2}}<BRbr>{{overline|5}} {{overline|2}}<BRbr>{{overline|6}} {{overline|2}}<BRbr>{{overline|n}} {{overline|2}}

| 222<BRbr>223<BRbr>224<BRbr>225<BRbr>226<BRbr>22n

| D<SUB>2</SUB><BRbr>D<SUB>3</SUB><BRbr>D<SUB>4</SUB><BRbr>D<SUB>5</SUB><BRbr>D<SUB>6</SUB><BRbr>D<SUB>n</SUB>

| D<SUB>4</SUB><BRbr>D<SUB>6</SUB><BRbr>D<SUB>8</SUB><BRbr>D<SUB>10</SUB><BRbr>D<SUB>12</SUB><BRbr>D<SUB>2n</SUB>

| [2,2]⁺<BRbr>[2,3]⁺<BRbr>[2,4]⁺<BRbr>[2,5]⁺<BRbr>[2,6]⁺<BRbr>[2,n]⁺

| 4<BRbr>6<BRbr>8<BRbr>10<BRbr>12<BRbr>2''n''

| mmm<BRbr>{{overline|6}}m2<BRbr>4/mmm<BRbr>{{overline|10}}m2<BRbr>6/mmm<br>n/mmm<br>{{overline|2n}}m2

| 2 2<BRbr>3 2<BRbr>4 2<BRbr>5 2<BRbr>6 2<BRbr>n 2

| *222<BRbr>*223<BRbr>*224<BRbr>*225<BRbr>*226<BRbr>*22n

| D<SUB>2h</SUB><BRbr>D<SUB>3h</SUB><BRbr>D<SUB>4h</SUB><BRbr>D<SUB>5h</SUB><BRbr>D<SUB>6h</SUB><BRbr>D<SUB>nh</SUB>

| ±D<SUB>4</SUB><BRbr>DD<SUB>12</SUB><BRbr>±D<SUB>8</SUB><BRbr>DD<SUB>20</SUB><BRbr>±D<SUB>12</SUB><BRbr>±D<SUB>2n</SUB> / DD<SUB>4n</SUB>

| [2,2]<BRbr>[2,3]<BRbr>[2,4]<BRbr>[2,5]<BRbr>[2,6]<BRbr>[2,n]

| 8<BRbr>12<BRbr>16<BRbr>20<BRbr>24<BRbr>4''n''

| {{Overline|4}}2m<BRbr>{{Overline|3}}m<BRbr>{{Overline|8}}2m<BRbr>{{Overline|5}}m<BRbr>{{Overline|12}}2m<br>{{Overline|2n}}2m<br>{{Overline|n}}m

| 4 {{overline|2}}<BRbr>6 {{overline|2}}<BRbr>8 {{overline|2}}<BRbr>10 {{overline|2}}<BRbr>12 {{overline|2}}<BRbr>n {{overline|2}}<BRbr>

| 2*2<BRbr>2*3<BRbr>2*4<BRbr>2*5<BRbr>2*6<BRbr>2*n

| D<SUB>2d</SUB><BRbr>D<SUB>3d</SUB><BRbr>D<SUB>4d</SUB><BRbr>D<SUB>5d</SUB><BRbr>D<SUB>6d</SUB><BRbr>D<SUB>nd</SUB>

| ±D<SUB>4</SUB><BRbr>±D<SUB>6</SUB><BRbr>DD<SUB>16</SUB><BRbr>±D<SUB>10</SUB><BRbr>DD<SUB>24</SUB><BRbr>DD<SUB>4n</SUB> / ±D<SUB>2n</SUB>

| [2⁺,4]<BRbr>[2⁺,6]<BRbr>[2⁺,8]<BRbr>[2⁺,10]<BRbr>[2⁺,12]<BRbr>[2⁺,2n]

| 8<BRbr>12<BRbr>16<BRbr>20<BRbr>24<BRbr>4''n''

! [[Notasi Schönflies | Schönflies]]

! colspan=3 | [[Diagram Coxeter–Dynkin | Diagram Coxeter]]

||{{CDD|node_c1|n|node_c2|2|node_c3}}||||4''n''||''n''-gonal [[Prisma (geometri) | prisma]]

!colspan=2|[[Coxeter&ndash;DynkinCoxeter–Dynkin diagram|Coxeter diagram]]

! colspan=2 | [[Coxeter-Dynkin diagram|Coxeter<BRbr>diagrams]]

* [[Harold Scott MacDonald Coxeter|H. S. M. Coxeter]]: ''Kaleidoscopes: Selected Writings of 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] {{Webarchive|url=https://web.archive.org/web/20160711140441/http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html |date=2016-07-11 }}

** (Paper 23) H. S. M. Coxeter, ''Regular and Semi-Regular Polytopes II'', [Math. Zeit. 188 (1985) 559&ndash;591559–591]

*[http://www.reciprocalnet.org/edumodules/symmetry/index.html Web-based point group tutorial] {{Webarchive|url=https://web.archive.org/web/20200222232344/http://www.reciprocalnet.org/edumodules/symmetry/index.html |date=2020-02-22 }} (needs Java and Flash)

*[http://plus.swap-zt.com/2010/06/sieve Subgroup enumeration] {{Webarchive|url=https://web.archive.org/web/20110824153712/http://plus.swap-zt.com/2010/06/sieve/ |date=2011-08-24 }} (needs Java)

* [http://www.geom.uiuc.edu/docs/reference/CRC-formulas/node9.html The Geometry Center: 2.1 Formulas for Symmetries in Cartesian Coordinates (two dimensions)] {{Webarchive|url=https://web.archive.org/web/20210418133146/http://www.geom.uiuc.edu/docs/reference/CRC-formulas/node9.html |date=2021-04-18 }}

* [http://www.geom.uiuc.edu/docs/reference/CRC-formulas/node45.html The Geometry Center: 10.1 Formulas for Symmetries in Cartesian Coordinates (three dimensions)] {{Webarchive|url=https://web.archive.org/web/20210418133153/http://www.geom.uiuc.edu/docs/reference/CRC-formulas/node45.html |date=2021-04-18 }}

[[Kategori: Kesimetrian EuklideanKristalografi]]

[[Kategori:Kesimetrian Teori grupEuklidean]]