Symmetry group (InstanceClass, 3)
From Hi.gher. Space
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== Three dimensions == | == Three dimensions == | ||
| - | The following table contains all the symmetry groups used by [[CRF polyhedra]]: | + | The following table contains all the symmetry groups used by [[CRF polyhedra]]. The order of each group is given after an asterisk: |
{| class='wikitable' | {| class='wikitable' | ||
!''n'' = !!2!!3!!4!!5 | !''n'' = !!2!!3!!4!!5 | ||
|- | |- | ||
!<br/>''special'' | !<br/>''special'' | ||
| - | |C<sub>s</sub><br/>[[Plane symmetry|Plane]] | + | |C<sub>s</sub> *2<br/>[[Plane symmetry|Plane]] |
| | | | ||
| | | | ||
| | | | ||
|- | |- | ||
| - | !C<sub>''n''</sub><br/>Chiral axial | + | !C<sub>''n''</sub> *''n''<br/>Chiral axial |
| | | | ||
| | | | ||
| | | | ||
| - | |C<sub>5</sub><br/>[[Chiral rhodoaxial symmetry|Chiral rhodoaxial]] | + | |C<sub>5</sub> *5<br/>[[Chiral rhodoaxial symmetry|Chiral rhodoaxial]] |
|- | |- | ||
| - | !C<sub>''n''v</sub><br/>Axial | + | !C<sub>''n''v</sub> *2''n''<br/>Axial |
| - | |C<sub>2v</sub><br/>[[Biaxial symmetry|Biaxial (biplane)]] | + | |C<sub>2v</sub> *4<br/>[[Biaxial symmetry|Biaxial (biplane)]] |
| - | |C<sub>3v</sub><br/>[[Pyroaxial symmetry|Pyroaxial]] | + | |C<sub>3v</sub> *6<br/>[[Pyroaxial symmetry|Pyroaxial]] |
| - | |C<sub>4v</sub><br/>[[Stauroaxial symmetry|Stauroaxial]] | + | |C<sub>4v</sub> *8<br/>[[Stauroaxial symmetry|Stauroaxial]] |
| - | |C<sub>5v</sub><br/>[[Rhodoaxial symmetry|Rhodoaxial]] | + | |C<sub>5v</sub> *10<br/>[[Rhodoaxial symmetry|Rhodoaxial]] |
|- | |- | ||
| - | !D<sub>''n''</sub><br/>Chiral antiprismatic | + | !D<sub>''n''</sub> *2''n''<br/>Chiral antiprismatic |
| | | | ||
| - | |D<sub>3</sub><br/>[[Chiral pyroantiprismatic symmetry|Chiral pyroantiprismatic]] | + | |D<sub>3</sub> *6<br/>[[Chiral pyroantiprismatic symmetry|Chiral pyroantiprismatic]] |
| - | |D<sub>4</sub><br/>[[Chiral stauroantiprismatic symmetry|Chiral stauroantiprismatic]] | + | |D<sub>4</sub> *8<br/>[[Chiral stauroantiprismatic symmetry|Chiral stauroantiprismatic]] |
| - | |D<sub>5</sub><br/>[[Chiral rhodoantiprismatic symmetry|Chiral rhodoantiprismatic]] | + | |D<sub>5</sub> *10<br/>[[Chiral rhodoantiprismatic symmetry|Chiral rhodoantiprismatic]] |
|- | |- | ||
| - | !D<sub>''n''d</sub><br/>Antiprismatic | + | !D<sub>''n''d</sub> *4''n''<br/>Antiprismatic |
| - | |D<sub>2d</sub><br/>[[Biantiprismatic symmetry|Biantiprismatic]] | + | |D<sub>2d</sub> *8<br/>[[Biantiprismatic symmetry|Biantiprismatic]] |
| - | |D<sub>3d</sub><br/>[[Pyroantiprismatic symmetry|Pyroantiprismatic]] | + | |D<sub>3d</sub> *12<br/>[[Pyroantiprismatic symmetry|Pyroantiprismatic]] |
| - | |D<sub>4d</sub><br/>[[Stauroantiprismatic symmetry|Stauroantiprismatic]] | + | |D<sub>4d</sub> *16<br/>[[Stauroantiprismatic symmetry|Stauroantiprismatic]] |
| - | |D<sub>5d</sub><br/>[[Rhodoantiprismatic symmetry|Rhodoantiprismatic]] | + | |D<sub>5d</sub> *20<br/>[[Rhodoantiprismatic symmetry|Rhodoantiprismatic]] |
|- | |- | ||
| - | !D<sub>''n''h</sub><br/>Prismatic | + | !D<sub>''n''h</sub> *4''n''<br/>Prismatic |
| - | |D<sub>2h</sub><br/>[[Biprismatic symmetry|Biprismatic (triplane, brick)]] | + | |D<sub>2h</sub> *8<br/>[[Biprismatic symmetry|Biprismatic (triplane, brick)]] |
| - | |D<sub>3h</sub><br/>[[Pyroprismatic symmetry|Pyroprismatic]] | + | |D<sub>3h</sub> *12<br/>[[Pyroprismatic symmetry|Pyroprismatic]] |
| - | |D<sub>4h</sub><br/>[[Stauroprismatic symmetry|Stauroprismatic]] | + | |D<sub>4h</sub> *16<br/>[[Stauroprismatic symmetry|Stauroprismatic]] |
| - | |D<sub>5h</sub><br/>[[Rhodoprismatic symmetry|Rhodoprismatic]] | + | |D<sub>5h</sub> *20<br/>[[Rhodoprismatic symmetry|Rhodoprismatic]] |
|- | |- | ||
!<br/>Chiral polyhedral | !<br/>Chiral polyhedral | ||
| | | | ||
| | | | ||
| - | |O<br/>[[Chiral staurohedral symmetry|Chiral staurohedral]] | + | |O *24<br/>[[Chiral staurohedral symmetry|Chiral staurohedral]] |
| - | |I<br/>[[Chiral rhodohedral symmetry|Chiral rhodohedral]] | + | |I *60<br/>[[Chiral rhodohedral symmetry|Chiral rhodohedral]] |
|- | |- | ||
!<br/>Polyhedral | !<br/>Polyhedral | ||
| | | | ||
| - | |T<sub>d</sub><br/>[[Pyrohedral symmetry|Pyrohedral]] | + | |T<sub>d</sub> *24<br/>[[Pyrohedral symmetry|Pyrohedral]] |
| - | |O<sub>h</sub><br/>[[Staurohedral symmetry|Staurohedral]] | + | |O<sub>h</sub> *48<br/>[[Staurohedral symmetry|Staurohedral]] |
| - | |I<sub>h</sub><br/>[[Rhodohedral symmetry|Rhodohedral]] | + | |I<sub>h</sub> *120<br/>[[Rhodohedral symmetry|Rhodohedral]] |
|} | |} | ||
| - | There are five more 3D symmetry groups, which are not seen in CRF polyhedra: C<sub>1</sub> ([[asymmetry]]) | + | There are five more 3D symmetry groups, which are not seen in CRF polyhedra: |
| + | *C<sub>1</sub> *1 ([[asymmetry]]) | ||
| + | *C<sub>''n''h</sub> *2''n'' ([[cyclic prismatic symmetry]]) | ||
| + | *S<sub>''n''</sub> *''n'' (for even ''n'' only) (S<sub>2</sub> being [[point symmetry]]) | ||
| + | *T *12 ([[chiral pyrohedral symmetry]]) | ||
| + | *T<sub>h</sub> *24 ([[pyritohedral symmetry]]). | ||
Latest revision as of 19:10, 12 March 2016
WIP
Three dimensions
The following table contains all the symmetry groups used by CRF polyhedra. The order of each group is given after an asterisk:
| n = | 2 | 3 | 4 | 5 |
|---|---|---|---|---|
special | Cs *2 Plane | |||
| Cn *n Chiral axial | C5 *5 Chiral rhodoaxial | |||
| Cnv *2n Axial | C2v *4 Biaxial (biplane) | C3v *6 Pyroaxial | C4v *8 Stauroaxial | C5v *10 Rhodoaxial |
| Dn *2n Chiral antiprismatic | D3 *6 Chiral pyroantiprismatic | D4 *8 Chiral stauroantiprismatic | D5 *10 Chiral rhodoantiprismatic | |
| Dnd *4n Antiprismatic | D2d *8 Biantiprismatic | D3d *12 Pyroantiprismatic | D4d *16 Stauroantiprismatic | D5d *20 Rhodoantiprismatic |
| Dnh *4n Prismatic | D2h *8 Biprismatic (triplane, brick) | D3h *12 Pyroprismatic | D4h *16 Stauroprismatic | D5h *20 Rhodoprismatic |
Chiral polyhedral | O *24 Chiral staurohedral | I *60 Chiral rhodohedral | ||
Polyhedral | Td *24 Pyrohedral | Oh *48 Staurohedral | Ih *120 Rhodohedral |
There are five more 3D symmetry groups, which are not seen in CRF polyhedra:
- C1 *1 (asymmetry)
- Cnh *2n (cyclic prismatic symmetry)
- Sn *n (for even n only) (S2 being point symmetry)
- T *12 (chiral pyrohedral symmetry)
- Th *24 (pyritohedral symmetry).
