Grup titik
![]() Bauhinia blakeana bunga di bendera wilayah Hong Kong memiliki simetri C 5 ; bintang di setiap kelopak memiliki simetri D 5 . |
![]() Yin dan Yang simbol memiliki geometri simetri C 2 dengan warna terbalik |
Dalam geometri, grup titik adalah grup geometris simetri (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 M yang mengubah titik x menjadi titik y :
- y = Mx
dimana asal adalah titik tetap. Elemen kelompok titik dapat berupa rotasi (determinan dari M = 1) atau yang lain 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 salah satu teorema Bieberbach. setiap jumlah dimensi hanya memiliki jumlah terbatas dari kelompok titik yang simetris di beberapa kisi atau kisi dengan nomor itu. Ini adalah grup titik kristalografi.
Grup titik kiral dan akiral, grup refleksi
[sunting | sunting sumber]Kelompok titik dapat diklasifikasikan ke dalam kelompok kiral (atau rotasi murni) dan kelompok akiral .[1] Gugus kiral adalah subgrup dari grup ortogonal khusus SO( d ): grup ini hanya berisi transformasi ortogonal yang mempertahankan orientasi, yaitu, determinan +1. Gugus akiral juga mengandung transformasi determinan −1. Dalam gugus akiral, transformasi yang mempertahankan orientasi membentuk subgrup (kiral) dari indeks 2.
Grup Coxeter Hingga atau kelompok refleksi adalah kelompok titik yang dihasilkan murni oleh sekumpulan cermin pantul yang melewati titik yang sama. Grup peringkat n Coxeter memiliki mirror n dan diwakili oleh diagram Coxeter-Dynkin. Notasi Coxeter menawarkan notasi tanda kurung yang setara dengan diagram Coxeter, dengan simbol markup untuk rotasi dan grup titik subsimetri lainnya. Kelompok refleksi harus akiral (kecuali untuk grup trivial yang hanya mengandung elemen identitas).
Daftar grup titik
[sunting | sunting sumber]Artikel ini sedang dalam perbaikan. CATATAN: Bagian ini terdapat masalah dan belum diterjemahkan. Untuk menghindari konflik penyuntingan, mohon jangan melakukan penyuntingan selama pesan ini ditampilkan. Halaman ini terakhir disunting oleh InternetArchiveBot (Kontrib • Log) 395 hari 359 menit lalu. |
Satu dimensi
[sunting | sunting sumber]Hanya ada dua grup titik satu dimensi, yaitu grup identitas dan kelompok refleksi.
Group | Coxeter | Diagram Coxeter | Urutan | Deskripsi |
---|---|---|---|---|
C1 | [ ]+ | 1 | Identity | |
D1 | [ ] | ![]() |
2 | Reflection group |
Dua dimensi
[sunting | sunting sumber]Grup titik dalam dua dimensi, terkadang disebut grup roset.
Mereka datang dalam dua keluarga yang tidak terbatas:
- Grup siklik Cn of n - grup rotasi lipat
- Grup dihedral Dn of n-kelompok rotasi dan refleksi lipat
Menerapkan teorema pembatasan kristalografi membatasi n ke nilai 1, 2, 3, 4, dan 6 untuk kedua famili, menghasilkan 10 kelompok.
Grup | Intl | Orbifold | Coxeter | Urutan | Deskripsi |
---|---|---|---|---|---|
Cn | n | n• | [n]+ | n | Siklik: n - rotasi lipat. Grup abstrak Z n , grup bilangan bulat di bawah penambahan modulo n . |
Dn | nm | *n• | [n] | 2n | Dihedral: siklik dengan refleksi. Grup abstrak Dihn, grup dihedral. |
![](http://upload.wikimedia.org/wikipedia/commons/thumb/d/dd/Coxeter_diagram_finite_rank2_correspondence.png/220px-Coxeter_diagram_finite_rank2_correspondence.png)
Himpunan bagian dari grup titik pantulan murni, yang ditentukan oleh 1 atau 2 mirror, juga dapat diberikan oleh grup Coxeter dan poligon terkait. Ini termasuk 5 kelompok kristalografi. Simetri kelompok pantulan dapat digandakan dengan isomorphism, memetakan kedua cermin satu sama lain dengan cermin membagi dua, menggandakan simetri.
Reflektif | Rotasi | Terkait poligon | |||||||
---|---|---|---|---|---|---|---|---|---|
Group | Grup Coxeter | Diagram Coxeter | Urutan | Subgrup | Coxeter | Urutan | |||
D1 | A1 | [ ] | ![]() |
![]() |
2 | C1 | []+ | 1 | Digon |
D2 | A12 | [2] | ![]() ![]() ![]() |
![]() ![]() ![]() |
4 | C2 | [2]+ | 2 | Persegi panjang |
D3 | A2 | [3] | ![]() ![]() ![]() |
![]() ![]() ![]() |
6 | C3 | [3]+ | 3 | Segitiga sama sisi |
D4 | BC2 | [4] | ![]() ![]() ![]() |
![]() ![]() ![]() |
8 | C4 | [4]+ | 4 | Persegi |
D5 | H2 | [5] | ![]() ![]() ![]() |
![]() ![]() ![]() |
10 | C5 | [5]+ | 5 | Segi lima biasa |
D6 | G2 | [6] | ![]() ![]() ![]() |
![]() ![]() ![]() |
12 | C6 | [6]+ | 6 | Segi enam biasa |
Dn | I2(n) | [n] | ![]() ![]() ![]() |
![]() ![]() ![]() |
2n | Cn | [n]+ | n | Poligon beraturan |
D2×2 | A12×2 | [[2]] = [4] | ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() |
8 | ||||
D3×2 | A2×2 | [[3]] = [6] | ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() |
12 | ||||
D4×2 | BC2×2 | [[4]] = [8] | ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() |
16 | ||||
D5×2 | H2×2 | [[5]] = [10] | ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() |
20 | ||||
D6×2 | G2×2 | [[6]] = [12] | ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() |
24 | ||||
Dn×2 | I2(n)×2 | [[n]] = [2n] | ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
4n |
Tiga dimensi
[sunting | sunting sumber]Grup titik dalam tiga dimensi, kadang disebut grup titik molekul setelah digunakan secara luas dalam mempelajari kesimetrian molekul kecil.
Mereka datang dalam 7 kelompok tak terbatas dari kelompok aksial atau prismatik, dan 7 kelompok polihedral atau Platonis tambahan. Dalam Notasi Schönflies, *
- Grup aksial: Cn, S2n, Cnh, Cnv, Dn, Dnd, Dnh
- Grup polihedral: T, Td, Th, O, Oh, I, Ih
Menerapkan teorema pembatasan kristalografi ke grup ini menghasilkan 32 grup titik kristalografi.
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(*) Ketika entri Intl digandakan, yang pertama untuk genap n , yang kedua untuk ganjil n . |
Grup refleksi
[sunting | sunting sumber]![](http://upload.wikimedia.org/wikipedia/commons/thumb/e/e0/Coxeter_diagram_finite_rank3_correspondence.png/220px-Coxeter_diagram_finite_rank3_correspondence.png)
Kelompok titik refleksi, ditentukan oleh 1 sampai 3 bidang cermin, juga dapat diberikan oleh grup Coxeter dan polihedra terkait. Grup [3,3] dapat digandakan, ditulis sebagai [[ 3,3]], memetakan cermin pertama dan terakhir satu sama lain, menggandakan simetri menjadi 48, dan isomorfik ke grup [4,3].
Schönflies | grup Coxeter | Diagram Coxeter | Urutan | Terkait reguler dan | |||
---|---|---|---|---|---|---|---|
Td | A3 | [3,3] | ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() |
24 | Tetrahedron | |
Td×Dih1 = Oh | A3×2 = BC3 | [[3,3]] = [4,3] | ![]() ![]() ![]() ![]() ![]() |
= ![]() ![]() ![]() ![]() ![]() |
48 | Oktahedron bintang | |
Oh | BC3 | [4,3] | ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() |
48 | kubus, segi delapan | |
Ih | H3 | [5,3] | ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() |
120 | Icosahedron, dodecahedron | |
D3h | A2×A1 | [3,2] | ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() |
12 | Prisma segitiga | |
D3h×Dih1 = D6h | A2×A1×2 | [[3],2] | ![]() ![]() ![]() ![]() ![]() |
= ![]() ![]() ![]() ![]() ![]() |
24 | Prisma segi enam | |
D4h | BC2×A1 | [4,2] | ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() |
16 | Prisma persegi | |
D4h×Dih1 = D8h | BC2×A1×2 | [[4],2] = [8,2] | ![]() ![]() ![]() ![]() ![]() |
= ![]() ![]() ![]() ![]() ![]() |
32 | Octagonal prism | |
D5h | H2×A1 | [5,2] | ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() |
20 | Prisma segi lima | |
D6h | G2×A1 | [6,2] | ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() |
24 | Prisma segi enam | |
Dnh | I2(n)×A1 | [n,2] | ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() |
4n | n-gonal prisma | |
Dnh×Dih1 = D2nh | I2(n)×A1×2 | [[n],2] | ![]() ![]() ![]() ![]() ![]() |
= ![]() ![]() ![]() ![]() ![]() ![]() |
8n | ||
D2h | A13 | [2,2] | ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() |
8 | Balok | |
D2h×Dih1 | A13×2 | [[2],2] = [4,2] | ![]() ![]() ![]() ![]() ![]() |
= ![]() ![]() ![]() ![]() ![]() |
16 | ||
D2h×Dih3 = Oh | A13×6 | [3[2,2]] = [4,3] | ![]() ![]() ![]() ![]() ![]() |
= ![]() ![]() ![]() ![]() ![]() |
48 | ||
C3v | A2 | [1,3] | ![]() ![]() ![]() |
![]() ![]() ![]() |
6 | Hosohedron | |
C4v | BC2 | [1,4] | ![]() ![]() ![]() |
![]() ![]() ![]() |
8 | ||
C5v | H2 | [1,5] | ![]() ![]() ![]() |
![]() ![]() ![]() |
10 | ||
C6v | G2 | [1,6] | ![]() ![]() ![]() |
![]() ![]() ![]() |
12 | ||
Cnv | I2(n) | [1,n] | ![]() ![]() ![]() |
![]() ![]() ![]() |
2n | ||
Cnv×Dih1 = C2nv | I2(n)×2 | [1,[n]] = [1,2n] | ![]() ![]() ![]() |
= ![]() ![]() ![]() ![]() |
4n | ||
C2v | A12 | [1,2] | ![]() ![]() ![]() |
![]() ![]() ![]() |
4 | ||
C2v×Dih1 | A12×2 | [1,[2]] | ![]() ![]() ![]() |
= ![]() ![]() ![]() |
8 | ||
Cs | A1 | [1,1] | ![]() |
![]() |
2 |
Four dimensions
[sunting | sunting sumber]The four-dimensional point groups (chiral as well as achiral) are listed in Conway and Smith,[1] Section 4, Tables 4.1-4.3.
![](http://upload.wikimedia.org/wikipedia/commons/thumb/4/46/Coxeter_diagram_finite_rank4_correspondence.png/220px-Coxeter_diagram_finite_rank4_correspondence.png)
The following list gives the four-dimensional reflection groups (excluding those that leave a subspace fixed and that are therefore lower-dimensional reflection groups). Each group is specified as a Coxeter group, and like the polyhedral groups of 3D, it can be named by its related convex regular 4-polytope. Related pure rotational groups exist for each with half the order, and can be represented by the bracket Coxeter notation with a '+' exponent, for example [3,3,3]+ has three 3-fold gyration points and symmetry order 60. Front-back symmetric groups like [3,3,3] and [3,4,3] can be doubled, shown as double brackets in Coxeter's notation, for example [[3,3,3]] with its order doubled to 240.
Coxeter group/notation | Coxeter diagram | Order | Related polytopes | ||
---|---|---|---|---|---|
A4 | [3,3,3] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
120 | 5-cell | |
A4×2 | [[3,3,3]] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
240 | 5-cell dual compound | |
BC4 | [4,3,3] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
384 | 16-cell/Tesseract | |
D4 | [31,1,1] | ![]() ![]() ![]() ![]() ![]() |
192 | Demitesseractic | |
D4×2 = BC4 | <[3,31,1]> = [4,3,3] | ![]() ![]() ![]() ![]() ![]() |
= ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
384 | |
D4×6 = F4 | [3[31,1,1]] = [3,4,3] | ![]() ![]() ![]() ![]() ![]() |
= ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
1152 | |
F4 | [3,4,3] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
1152 | 24-cell | |
F4×2 | [[3,4,3]] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
2304 | 24-cell dual compound | |
H4 | [5,3,3] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
14400 | 120-cell/600-cell | |
A3×A1 | [3,3,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
48 | Tetrahedral prism | |
A3×A1×2 | [[3,3],2] = [4,3,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
= ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
96 | Octahedral prism |
BC3×A1 | [4,3,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
96 | ||
H3×A1 | [5,3,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
240 | Icosahedral prism | |
A2×A2 | [3,2,3] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
36 | Duoprism | |
A2×BC2 | [3,2,4] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
48 | ||
A2×H2 | [3,2,5] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
60 | ||
A2×G2 | [3,2,6] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
72 | ||
BC2×BC2 | [4,2,4] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
64 | ||
BC22×2 | [[4,2,4]] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
128 | ||
BC2×H2 | [4,2,5] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
80 | ||
BC2×G2 | [4,2,6] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
96 | ||
H2×H2 | [5,2,5] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
100 | ||
H2×G2 | [5,2,6] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
120 | ||
G2×G2 | [6,2,6] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
144 | ||
I2(p)×I2(q) | [p,2,q] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
4pq | ||
I2(2p)×I2(q) | [[p],2,q] = [2p,2,q] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
= ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
8pq | |
I2(2p)×I2(2q) | [[p]],2,[[q]] = [2p,2,2q] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
= ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
16pq | |
I2(p)2×2 | [[p,2,p]] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
8p2 | ||
I2(2p)2×2 | [[[p],2,[p]]] = [[2p,2,2p]] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
= ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
32p2 | |
A2×A1×A1 | [3,2,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
24 | ||
BC2×A1×A1 | [4,2,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
32 | ||
H2×A1×A1 | [5,2,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
40 | ||
G2×A1×A1 | [6,2,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
48 | ||
I2(p)×A1×A1 | [p,2,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
8p | ||
I2(2p)×A1×A1×2 | [[p],2,2] = [2p,2,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
= ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
16p | |
I2(p)×A12×2 | [p,2,[2]] = [p,2,4] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
= ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
16p | |
I2(2p)×A12×4 | [[p]],2,[[2]] = [2p,2,4] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
= ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
32p | |
A1×A1×A1×A1 | [2,2,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
16 | 4-orthotope | |
A12×A1×A1×2 | [[2],2,2] = [4,2,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
= ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
32 | |
A12×A12×4 | [[2]],2,[[2]] = [4,2,4] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
= ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
64 | |
A13×A1×6 | [3[2,2],2] = [4,3,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
= ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
96 | |
A14×24 | [3,3[2,2,2]] = [4,3,3] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
= ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
384 |
Five dimensions
[sunting | sunting sumber]![](http://upload.wikimedia.org/wikipedia/commons/thumb/e/e0/Coxeter_diagram_finite_rank5_correspondence.png/220px-Coxeter_diagram_finite_rank5_correspondence.png)
The following table gives the five-dimensional reflection groups (excluding those that are lower-dimensional reflection groups), by listing them as Coxeter groups. Related chiral groups exist for each with half the order, and can be represented by the bracket Coxeter notation with a '+' exponent, for example [3,3,3,3]+ has four 3-fold gyration points and symmetry order 360.
Coxeter group/notation | Coxeter diagrams |
Order | Related regular and prismatic polytopes | ||
---|---|---|---|---|---|
A5 | [3,3,3,3] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
720 | 5-simplex |
A5×2 | [[3,3,3,3]] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
1440 | 5-simplex dual compound |
BC5 | [4,3,3,3] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
3840 | 5-cube, 5-orthoplex |
D5 | [32,1,1] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
1920 | 5-demicube |
D5×2 | <[3,3,31,1]> | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
3840 | |
A4×A1 | [3,3,3,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
240 | 5-cell prism |
A4×A1×2 | [[3,3,3],2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
480 | |
BC4×A1 | [4,3,3,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
768 | tesseract prism |
F4×A1 | [3,4,3,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
2304 | 24-cell prism |
F4×A1×2 | [[3,4,3],2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
4608 | |
H4×A1 | [5,3,3,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
28800 | 600-cell or 120-cell prism |
D4×A1 | [31,1,1,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
384 | Demitesseract prism |
A3×A2 | [3,3,2,3] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
144 | Duoprism |
A3×A2×2 | [[3,3],2,3] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
288 | |
A3×BC2 | [3,3,2,4] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
192 | |
A3×H2 | [3,3,2,5] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
240 | |
A3×G2 | [3,3,2,6] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
288 | |
A3×I2(p) | [3,3,2,p] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
48p | |
BC3×A2 | [4,3,2,3] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
288 | |
BC3×BC2 | [4,3,2,4] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
384 | |
BC3×H2 | [4,3,2,5] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
480 | |
BC3×G2 | [4,3,2,6] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
576 | |
BC3×I2(p) | [4,3,2,p] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
96p | |
H3×A2 | [5,3,2,3] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
720 | |
H3×BC2 | [5,3,2,4] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
960 | |
H3×H2 | [5,3,2,5] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
1200 | |
H3×G2 | [5,3,2,6] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
1440 | |
H3×I2(p) | [5,3,2,p] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
240p | ||
A3×A12 | [3,3,2,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
96 | ||
BC3×A12 | [4,3,2,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
192 | ||
H3×A12 | [5,3,2,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
480 | ||
A22×A1 | [3,2,3,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
72 | duoprism prism | |
A2×BC2×A1 | [3,2,4,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
96 | ||
A2×H2×A1 | [3,2,5,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
120 | ||
A2×G2×A1 | [3,2,6,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
144 | ||
BC22×A1 | [4,2,4,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
128 | ||
BC2×H2×A1 | [4,2,5,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
160 | ||
BC2×G2×A1 | [4,2,6,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
192 | ||
H22×A1 | [5,2,5,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
200 | ||
H2×G2×A1 | [5,2,6,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
240 | ||
G22×A1 | [6,2,6,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
288 | ||
I2(p)×I2(q)×A1 | [p,2,q,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
8pq | ||
A2×A13 | [3,2,2,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
48 | ||
BC2×A13 | [4,2,2,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
64 | ||
H2×A13 | [5,2,2,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
80 | ||
G2×A13 | [6,2,2,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
96 | ||
I2(p)×A13 | [p,2,2,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
16p | ||
A15 | [2,2,2,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
32 | 5-orthotope |
A15×(2!) | [[2],2,2,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
64 | |
A15×(2!×2!) | [[2]],2,[2],2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
128 | |
A15×(3!) | [3[2,2],2,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
192 | |
A15×(3!×2!) | [3[2,2],2,[[2]] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
384 | |
A15×(4!) | [3,3[2,2,2],2]] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
768 | |
A15×(5!) | [3,3,3[2,2,2,2]] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
3840 |
Six dimensions
[sunting | sunting sumber]![](http://upload.wikimedia.org/wikipedia/commons/thumb/5/5d/Coxeter_diagram_finite_rank6_correspondence.png/220px-Coxeter_diagram_finite_rank6_correspondence.png)
The following table gives the six-dimensional reflection groups (excluding those that are lower-dimensional reflection groups), by listing them as Coxeter groups. Related pure rotational groups exist for each with half the order, and can be represented by the bracket Coxeter notation with a '+' exponent, for example [3,3,3,3,3]+ has five 3-fold gyration points and symmetry order 2520.
Coxeter group | Coxeter diagram |
Order | Related regular and prismatic polytopes | |
---|---|---|---|---|
A6 | [3,3,3,3,3] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
5040 (7!) | 6-simplex |
A6×2 | [[3,3,3,3,3]] | ![]() ![]() ![]() ![]() ![]() |
10080 (2×7!) | 6-simplex dual compound |
BC6 | [4,3,3,3,3] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
46080 (26×6!) | 6-cube, 6-orthoplex |
D6 | [3,3,3,31,1] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
23040 (25×6!) | 6-demicube |
E6 | [3,32,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
51840 (72×6!) | 122, 221 |
A5×A1 | [3,3,3,3,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
1440 (2×6!) | 5-simplex prism |
BC5×A1 | [4,3,3,3,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
7680 (26×5!) | 5-cube prism |
D5×A1 | [3,3,31,1,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
3840 (25×5!) | 5-demicube prism |
A4×I2(p) | [3,3,3,2,p] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
240p | Duoprism |
BC4×I2(p) | [4,3,3,2,p] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
768p | |
F4×I2(p) | [3,4,3,2,p] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
2304p | |
H4×I2(p) | [5,3,3,2,p] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
28800p | |
D4×I2(p) | [3,31,1,2,p] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
384p | |
A4×A12 | [3,3,3,2,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
480 | |
BC4×A12 | [4,3,3,2,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
1536 | |
F4×A12 | [3,4,3,2,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
4608 | |
H4×A12 | [5,3,3,2,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
57600 | |
D4×A12 | [3,31,1,2,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
768 | |
A32 | [3,3,2,3,3] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
576 | |
A3×BC3 | [3,3,2,4,3] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
1152 | |
A3×H3 | [3,3,2,5,3] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
2880 | |
BC32 | [4,3,2,4,3] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
2304 | |
BC3×H3 | [4,3,2,5,3] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
5760 | |
H32 | [5,3,2,5,3] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
14400 | |
A3×I2(p)×A1 | [3,3,2,p,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
96p | Duoprism prism |
BC3×I2(p)×A1 | [4,3,2,p,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
192p | |
H3×I2(p)×A1 | [5,3,2,p,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
480p | |
A3×A13 | [3,3,2,2,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
192 | |
BC3×A13 | [4,3,2,2,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
384 | |
H3×A13 | [5,3,2,2,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
960 | |
I2(p)×I2(q)×I2(r) | [p,2,q,2,r] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
8pqr | Triaprism |
I2(p)×I2(q)×A12 | [p,2,q,2,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
16pq | |
I2(p)×A14 | [p,2,2,2,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
32p | |
A16 | [2,2,2,2,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
64 | 6-orthotope |
Seven dimensions
[sunting | sunting sumber]The following table gives the seven-dimensional reflection groups (excluding those that are lower-dimensional reflection groups), by listing them as Coxeter groups. Related chiral groups exist for each with half the order, defined by an even number of reflections, and can be represented by the bracket Coxeter notation with a '+' exponent, for example [3,3,3,3,3,3]+ has six 3-fold gyration points and symmetry order 20160.
Coxeter group | Coxeter diagram | Order | Related polytopes | |
---|---|---|---|---|
A7 | [3,3,3,3,3,3] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
40320 (8!) | 7-simplex |
A7×2 | [[3,3,3,3,3,3]] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
80640 (2×8!) | 7-simplex dual compound |
BC7 | [4,3,3,3,3,3] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
645120 (27×7!) | 7-cube, 7-orthoplex |
D7 | [3,3,3,3,31,1] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
322560 (26×7!) | 7-demicube |
E7 | [3,3,3,32,1] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
2903040 (8×9!) | 321, 231, 132 |
A6×A1 | [3,3,3,3,3,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
10080 (2×7!) | |
BC6×A1 | [4,3,3,3,3,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
92160 (27×6!) | |
D6×A1 | [3,3,3,31,1,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
46080 (26×6!) | |
E6×A1 | [3,3,32,1,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
103680 (144×6!) | |
A5×I2(p) | [3,3,3,3,2,p] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
1440p | |
BC5×I2(p) | [4,3,3,3,2,p] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
7680p | |
D5×I2(p) | [3,3,31,1,2,p] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
3840p | |
A5×A12 | [3,3,3,3,2,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
2880 | |
BC5×A12 | [4,3,3,3,2,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
15360 | |
D5×A12 | [3,3,31,1,2,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
7680 | |
A4×A3 | [3,3,3,2,3,3] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
2880 | |
A4×BC3 | [3,3,3,2,4,3] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
5760 | |
A4×H3 | [3,3,3,2,5,3] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
14400 | |
BC4×A3 | [4,3,3,2,3,3] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
9216 | |
BC4×BC3 | [4,3,3,2,4,3] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
18432 | |
BC4×H3 | [4,3,3,2,5,3] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
46080 | |
H4×A3 | [5,3,3,2,3,3] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
345600 | |
H4×BC3 | [5,3,3,2,4,3] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
691200 | |
H4×H3 | [5,3,3,2,5,3] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
1728000 | |
F4×A3 | [3,4,3,2,3,3] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
27648 | |
F4×BC3 | [3,4,3,2,4,3] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
55296 | |
F4×H3 | [3,4,3,2,5,3] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
138240 | |
D4×A3 | [31,1,1,2,3,3] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
4608 | |
D4×BC3 | [3,31,1,2,4,3] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
9216 | |
D4×H3 | [3,31,1,2,5,3] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
23040 | |
A4×I2(p)×A1 | [3,3,3,2,p,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
480p | |
BC4×I2(p)×A1 | [4,3,3,2,p,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
1536p | |
D4×I2(p)×A1 | [3,31,1,2,p,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
768p | |
F4×I2(p)×A1 | [3,4,3,2,p,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
4608p | |
H4×I2(p)×A1 | [5,3,3,2,p,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
57600p | |
A4×A13 | [3,3,3,2,2,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
960 | |
BC4×A13 | [4,3,3,2,2,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
3072 | |
F4×A13 | [3,4,3,2,2,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
9216 | |
H4×A13 | [5,3,3,2,2,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
115200 | |
D4×A13 | [3,31,1,2,2,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
1536 | |
A32×A1 | [3,3,2,3,3,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
1152 | |
A3×BC3×A1 | [3,3,2,4,3,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
2304 | |
A3×H3×A1 | [3,3,2,5,3,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
5760 | |
BC32×A1 | [4,3,2,4,3,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
4608 | |
BC3×H3×A1 | [4,3,2,5,3,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
11520 | |
H32×A1 | [5,3,2,5,3,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
28800 | |
A3×I2(p)×I2(q) | [3,3,2,p,2,q] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
96pq | |
BC3×I2(p)×I2(q) | [4,3,2,p,2,q] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
192pq | |
H3×I2(p)×I2(q) | [5,3,2,p,2,q] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
480pq | |
A3×I2(p)×A12 | [3,3,2,p,2,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
192p | |
BC3×I2(p)×A12 | [4,3,2,p,2,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
384p | |
H3×I2(p)×A12 | [5,3,2,p,2,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
960p | |
A3×A14 | [3,3,2,2,2,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
384 | |
BC3×A14 | [4,3,2,2,2,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
768 | |
H3×A14 | [5,3,2,2,2,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
1920 | |
I2(p)×I2(q)×I2(r)×A1 | [p,2,q,2,r,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
16pqr | |
I2(p)×I2(q)×A13 | [p,2,q,2,2,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
32pq | |
I2(p)×A15 | [p,2,2,2,2,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
64p | |
A17 | [2,2,2,2,2,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
128 |
Eight dimensions
[sunting | sunting sumber]The following table gives the eight-dimensional reflection groups (excluding those that are lower-dimensional reflection groups), by listing them as Coxeter groups. Related chiral groups exist for each with half the order, defined by an even number of reflections, and can be represented by the bracket Coxeter notation with a '+' exponent, for example [3,3,3,3,3,3,3]+ has seven 3-fold gyration points and symmetry order 181440.
Coxeter group | Coxeter diagram | Order | Related polytopes | |
---|---|---|---|---|
A8 | [3,3,3,3,3,3,3] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
362880 (9!) | 8-simplex |
A8×2 | [[3,3,3,3,3,3,3]] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
725760 (2×9!) | 8-simplex dual compound |
BC8 | [4,3,3,3,3,3,3] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
10321920 (288!) | 8-cube,8-orthoplex |
D8 | [3,3,3,3,3,31,1] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
5160960 (278!) | 8-demicube |
E8 | [3,3,3,3,32,1] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
696729600 (192×10!) | 421, 241, 142 |
A7×A1 | [3,3,3,3,3,3,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
80640 | 7-simplex prism |
BC7×A1 | [4,3,3,3,3,3,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
645120 | 7-cube prism |
D7×A1 | [3,3,3,3,31,1,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
322560 | 7-demicube prism |
E7×A1 | [3,3,3,32,1,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
5806080 | 321 prism, 231 prism, 142 prism |
A6×I2(p) | [3,3,3,3,3,2,p] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
10080p | duoprism |
BC6×I2(p) | [4,3,3,3,3,2,p] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
92160p | |
D6×I2(p) | [3,3,3,31,1,2,p] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
46080p | |
E6×I2(p) | [3,3,32,1,2,p] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
103680p | |
A6×A12 | [3,3,3,3,3,2,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
20160 | |
BC6×A12 | [4,3,3,3,3,2,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
184320 | |
D6×A12 | [33,1,1,2,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
92160 | |
E6×A12 | [3,3,32,1,2,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
207360 | |
A5×A3 | [3,3,3,3,2,3,3] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
17280 | |
BC5×A3 | [4,3,3,3,2,3,3] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
92160 | |
D5×A3 | [32,1,1,2,3,3] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
46080 | |
A5×BC3 | [3,3,3,3,2,4,3] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
34560 | |
BC5×BC3 | [4,3,3,3,2,4,3] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
184320 | |
D5×BC3 | [32,1,1,2,4,3] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
92160 | |
A5×H3 | [3,3,3,3,2,5,3] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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BC5×H3 | [4,3,3,3,2,5,3] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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D5×H3 | [32,1,1,2,5,3] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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A5×I2(p)×A1 | [3,3,3,3,2,p,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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BC5×I2(p)×A1 | [4,3,3,3,2,p,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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D5×I2(p)×A1 | [32,1,1,2,p,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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A5×A13 | [3,3,3,3,2,2,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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BC5×A13 | [4,3,3,3,2,2,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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D5×A13 | [32,1,1,2,2,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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A4×A4 | [3,3,3,2,3,3,3] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
||
BC4×A4 | [4,3,3,2,3,3,3] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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D4×A4 | [31,1,1,2,3,3,3] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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F4×A4 | [3,4,3,2,3,3,3] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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H4×A4 | [5,3,3,2,3,3,3] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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BC4×BC4 | [4,3,3,2,4,3,3] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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D4×BC4 | [31,1,1,2,4,3,3] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
||
F4×BC4 | [3,4,3,2,4,3,3] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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H4×BC4 | [5,3,3,2,4,3,3] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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D4×D4 | [31,1,1,2,31,1,1] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
||
F4×D4 | [3,4,3,2,31,1,1] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
||
H4×D4 | [5,3,3,2,31,1,1] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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F4×F4 | [3,4,3,2,3,4,3] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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H4×F4 | [5,3,3,2,3,4,3] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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H4×H4 | [5,3,3,2,5,3,3] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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A4×A3×A1 | [3,3,3,2,3,3,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
duoprism prisms | |
A4×BC3×A1 | [3,3,3,2,4,3,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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A4×H3×A1 | [3,3,3,2,5,3,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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BC4×A3×A1 | [4,3,3,2,3,3,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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BC4×BC3×A1 | [4,3,3,2,4,3,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
||
BC4×H3×A1 | [4,3,3,2,5,3,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
||
H4×A3×A1 | [5,3,3,2,3,3,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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H4×BC3×A1 | [5,3,3,2,4,3,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
||
H4×H3×A1 | [5,3,3,2,5,3,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
||
F4×A3×A1 | [3,4,3,2,3,3,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
||
F4×BC3×A1 | [3,4,3,2,4,3,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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F4×H3×A1 | [3,4,2,3,5,3,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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D4×A3×A1 | [31,1,1,2,3,3,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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D4×BC3×A1 | [31,1,1,2,4,3,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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D4×H3×A1 | [31,1,1,2,5,3,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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A4×I2(p)×I2(q) | [3,3,3,2,p,2,q] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
triaprism | |
BC4×I2(p)×I2(q) | [4,3,3,2,p,2,q] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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F4×I2(p)×I2(q) | [3,4,3,2,p,2,q] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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H4×I2(p)×I2(q) | [5,3,3,2,p,2,q] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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D4×I2(p)×I2(q) | [31,1,1,2,p,2,q] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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A4×I2(p)×A12 | [3,3,3,2,p,2,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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BC4×I2(p)×A12 | [4,3,3,2,p,2,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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F4×I2(p)×A12 | [3,4,3,2,p,2,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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H4×I2(p)×A12 | [5,3,3,2,p,2,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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D4×I2(p)×A12 | [31,1,1,2,p,2,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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A4×A14 | [3,3,3,2,2,2,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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BC4×A14 | [4,3,3,2,2,2,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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F4×A14 | [3,4,3,2,2,2,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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H4×A14 | [5,3,3,2,2,2,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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D4×A14 | [31,1,1,2,2,2,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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A3×A3×I2(p) | [3,3,2,3,3,2,p] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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BC3×A3×I2(p) | [4,3,2,3,3,2,p] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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H3×A3×I2(p) | [5,3,2,3,3,2,p] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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BC3×BC3×I2(p) | [4,3,2,4,3,2,p] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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H3×BC3×I2(p) | [5,3,2,4,3,2,p] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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H3×H3×I2(p) | [5,3,2,5,3,2,p] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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A3×A3×A12 | [3,3,2,3,3,2,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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BC3×A3×A12 | [4,3,2,3,3,2,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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H3×A3×A12 | [5,3,2,3,3,2,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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BC3×BC3×A12 | [4,3,2,4,3,2,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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H3×BC3×A12 | [5,3,2,4,3,2,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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H3×H3×A12 | [5,3,2,5,3,2,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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A3×I2(p)×I2(q)×A1 | [3,3,2,p,2,q,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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BC3×I2(p)×I2(q)×A1 | [4,3,2,p,2,q,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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H3×I2(p)×I2(q)×A1 | [5,3,2,p,2,q,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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A3×I2(p)×A13 | [3,3,2,p,2,2,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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BC3×I2(p)×A13 | [4,3,2,p,2,2,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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H3×I2(p)×A13 | [5,3,2,p,2,2,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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A3×A15 | [3,3,2,2,2,2,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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BC3×A15 | [4,3,2,2,2,2,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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H3×A15 | [5,3,2,2,2,2,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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I2(p)×I2(q)×I2(r)×I2(s) | [p,2,q,2,r,2,s] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
16pqrs | |
I2(p)×I2(q)×I2(r)×A12 | [p,2,q,2,r,2,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
32pqr | |
I2(p)×I2(q)×A14 | [p,2,q,2,2,2,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
64pq | |
I2(p)×A16 | [p,2,2,2,2,2,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
128p | |
A18 | [2,2,2,2,2,2,2] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
256 |
Lihat pula
[sunting | sunting sumber]- Grup titik dalam dua dimensi
- Grup titik dalam tiga dimensi
- Gruo titik dalam empat dimensi
- Kristalografi
- Grup titik kristalografi
- Simetri molekul
- Grup luar angkasa
- Difraksi sinar-X
- Kisi Bravais
- Spektroskopi inframerah dari logam karbonil
Referensi
[sunting | sunting sumber]- ^ a b Conway, John H.; Smith, Derek A. (2003). On quaternions and octonions: their geometry, arithmetic, and symmetry. A K Peters. ISBN 978-1-56881-134-5.
- ^ Grup Ruang Kristalografi dalam aljabar geometris, D. Hestenes and J. Holt, Journal of Mathematical Physics. 48, 023514 (2007) (22 pages) PDF [1] Diarsipkan 2020-10-20 di Wayback Machine.
- 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 [2] Diarsipkan 2016-07-11 di Wayback Machine.
- (Paper 23) H. S. M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559–591]
- H. S. M. Coxeter and W. O. J. Moser. Generators and Relations for Discrete Groups 4th ed, Springer-Verlag. New York. 1980
- N. W. Johnson: Geometries and Transformations, (2018) Chapter 11: Finite symmetry groups
Pranala luar
[sunting | sunting sumber]- Web-based point group tutorial Diarsipkan 2020-02-22 di Wayback Machine. (needs Java and Flash)
- Subgroup enumeration Diarsipkan 2011-08-24 di Wayback Machine. (needs Java)
- The Geometry Center: 2.1 Formulas for Symmetries in Cartesian Coordinates (two dimensions) Diarsipkan 2021-04-18 di Wayback Machine.
- The Geometry Center: 10.1 Formulas for Symmetries in Cartesian Coordinates (three dimensions) Diarsipkan 2021-04-18 di Wayback Machine.