Editing Crystal system (section)

==Crystal classes==
{{main|Crystallographic point group}}
The 7 crystal systems consist of 32 crystal classes (corresponding to the 32 crystallographic point groups) as shown in the following table below:

{| class=wikitable
|-
! Crystal family
! Crystal system
! [[Point group]] / Crystal class
! [[Schönflies notation|Schönflies]]
! [[Hermann–Mauguin notation|Hermann–Mauguin]]
! [[Orbifold notation|Orbifold]]
! [[Coxeter notation|Coxeter]]
! Point symmetry
! [[Symmetry number|Order]]
! [[Group theory#Abstract groups|Abstract group]]
|-
! rowspan=2 colspan=2| [[triclinic crystal system|triclinic]]
| pedial	
| C<sub>1</sub>
| 1
| 11
| [&nbsp;]<sup>+</sup>
| [[Chirality (chemistry)|enantiomorphic]] [[Polar point group|polar]]
| 1
| trivial <math>\mathbb{Z}_1</math>
|-
| pinacoidal
| C<sub>i</sub> (S<sub>2</sub>)
| {{overline|1}}
| 1x
| [2,1<sup>+</sup>]
| [[centrosymmetric]]
| 2
| [[Cyclic group|cyclic]] <math>\mathbb{Z}_2</math>
|-
! rowspan=3 colspan=2 | [[monoclinic crystal system|monoclinic]]
| sphenoidal
| C<sub>2</sub>
| 2
| 22	
| [2,2]<sup>+</sup>
| [[Chirality (chemistry)|enantiomorphic]] [[Polar point group|polar]]
| 2
| [[Cyclic group|cyclic]] <math>\mathbb{Z}_2</math>
|-
| domatic
| C<sub>s</sub> (C<sub>1h</sub>)
| m
| *11
| [&nbsp;]
| [[Polar point group|polar]]
| 2
| [[Cyclic group|cyclic]] <math>\mathbb{Z}_2</math>
|-
| [[prism (geometry)|prismatic]]
| C<sub>2h</sub>	
| 2/m
| 2*
| [2,2<sup>+</sup>]
| [[centrosymmetric]]
| 4
| [[Klein four-group|Klein four]] <math>\mathbb{V} = \mathbb{Z}_2\times\mathbb{Z}_2</math>
|-
! rowspan=3 colspan=2| [[orthorhombic crystal system|orthorhombic]]
| rhombic-disphenoidal
| D<sub>2</sub> (V)
| 222
| 222	
| [2,2]<sup>+</sup>
| [[Chirality (chemistry)|enantiomorphic]]
| 4
| [[Klein four-group|Klein four]] <math>\mathbb{V} = \mathbb{Z}_2\times\mathbb{Z}_2</math>
|-
| rhombic-[[Pyramid (geometry)|pyramidal]]
| C<sub>2v</sub>	
| mm2
| *22
| [2]
| [[Polar point group|polar]]
| 4
| [[Klein four-group|Klein four]] <math>\mathbb{V} = \mathbb{Z}_2\times\mathbb{Z}_2</math>
|-
| rhombic-[[dipyramid]]al
| D<sub>2h</sub> (V<sub>h</sub>)	
| mmm
| *222
| [2,2]
| [[centrosymmetric]]
| 8
| <math>\mathbb{V}\times\mathbb{Z}_2</math>
|-
! rowspan=7 colspan=2| [[tetragonal crystal system|tetragonal]]
| tetragonal-pyramidal	
| C<sub>4</sub>
| 4
| 44
| [4]<sup>+</sup>
| [[Chirality (chemistry)|enantiomorphic]] [[Polar point group|polar]]
| 4
| [[Cyclic group|cyclic]] <math>\mathbb{Z}_4</math>
|-
| tetragonal-disphenoidal
| S<sub>4</sub>	
| {{overline|4}}
| 2x
| [2<sup>+</sup>,2]
| [[non-centrosymmetric]]
| 4
| [[Cyclic group|cyclic]] <math>\mathbb{Z}_4</math>
|-
| tetragonal-dipyramidal
| C<sub>4h</sub>	
| 4/m
| 4*
| [2,4<sup>+</sup>]
| [[centrosymmetric]]
| 8
| <math>\mathbb{Z}_4\times\mathbb{Z}_2</math>
|-
| tetragonal-trapezohedral
| D<sub>4</sub>	
| 422
| 422
| [2,4]<sup>+</sup>
| [[Chirality (chemistry)|enantiomorphic]]
| 8
| [[Dihedral group|dihedral]] <math>\mathbb{D}_8 = \mathbb{Z}_4\rtimes\mathbb{Z}_2</math>
|-
| ditetragonal-pyramidal
| C<sub>4v</sub>	
| 4mm
| *44
| [4]
| [[Polar point group|polar]]
| 8
| [[Dihedral group|dihedral]] <math>\mathbb{D}_8 = \mathbb{Z}_4\rtimes\mathbb{Z}_2</math>
|-
| tetragonal-scalenohedral
| D<sub>2d</sub> (V<sub>d</sub>)
| {{overline|4}}2m or {{overline|4}}m2
| 2*2
| [2<sup>+</sup>,4]
| [[non-centrosymmetric]]
| 8
| [[Dihedral group|dihedral]] <math>\mathbb{D}_8 = \mathbb{Z}_4\rtimes\mathbb{Z}_2</math>
|-
| ditetragonal-dipyramidal
| D<sub>4h</sub>	
| 4/mmm
| *422
| [2,4]
| [[centrosymmetric]]
| 16
| <math>\mathbb{D}_8\times\mathbb{Z}_2</math>
|-
! rowspan=12| [[hexagonal crystal family|hexagonal]]
! rowspan=5| trigonal 
| trigonal-pyramidal	
| C<sub>3</sub>
| 3
| 33
| [3]<sup>+</sup>
| [[Chirality (chemistry)|enantiomorphic]] [[Polar point group|polar]]
| 3
| [[Cyclic group|cyclic]] <math>\mathbb{Z}_3</math>
|-
| rhombohedral
| C<sub>3i</sub> (S<sub>6</sub>)
| {{overline|3}}
| 3x
| [2<sup>+</sup>,3<sup>+</sup>]
| [[centrosymmetric]]
| 6
| [[Cyclic group|cyclic]] <math>\mathbb{Z}_6 = \mathbb{Z}_3\times\mathbb{Z}_2</math>
|-
| trigonal-trapezohedral
| D<sub>3</sub>	
| 32 or 321 or 312
| 322
| [3,2]<sup>+</sup>
| [[Chirality (chemistry)|enantiomorphic]]
| 6
| [[Dihedral group|dihedral]] <math>\mathbb{D}_6 = \mathbb{Z}_3\rtimes\mathbb{Z}_2</math>
|-
| ditrigonal-pyramidal
| C<sub>3v</sub>	
| 3m or 3m1 or 31m
| *33
| [3]
| [[Polar point group|polar]]
| 6
| [[Dihedral group|dihedral]] <math>\mathbb{D}_6 = \mathbb{Z}_3\rtimes\mathbb{Z}_2</math>
|-
| ditrigonal-scalenohedral
| D<sub>3d</sub>	
| {{overline|3}}m or {{overline|3}}m1 or {{overline|3}}1m
| 2*3
| [2<sup>+</sup>,6]
| [[centrosymmetric]]
| 12
| [[Dihedral group|dihedral]] <math>\mathbb{D}_{12} = \mathbb{Z}_6\rtimes\mathbb{Z}_2</math>
|-
! rowspan=7 | hexagonal
| hexagonal-pyramidal	
| C<sub>6</sub>
| 6
| 66
| [6]<sup>+</sup>
| [[Chirality (chemistry)|enantiomorphic]] [[Polar point group|polar]]
| 6
| [[Cyclic group|cyclic]] <math>\mathbb{Z}_6 = \mathbb{Z}_3\times\mathbb{Z}_2</math>
|-
| trigonal-dipyramidal
| C<sub>3h</sub>
| {{overline|6}}
| 3*
| [2,3<sup>+</sup>]
| [[non-centrosymmetric]]
| 6
| [[Cyclic group|cyclic]] <math>\mathbb{Z}_6 = \mathbb{Z}_3\times\mathbb{Z}_2</math>
|-
| hexagonal-dipyramidal
| C<sub>6h</sub>	
| 6/m
| 6*
| [2,6<sup>+</sup>]
| [[centrosymmetric]]
| 12
| <math>\mathbb{Z}_6\times\mathbb{Z}_2</math>
|-
| hexagonal-trapezohedral
| D<sub>6</sub>	
| 622
| 622
| [2,6]<sup>+</sup>
| [[Chirality (chemistry)|enantiomorphic]]
| 12
| [[Dihedral group|dihedral]] <math>\mathbb{D}_{12} = \mathbb{Z}_6\rtimes\mathbb{Z}_2</math>
|-
| dihexagonal-pyramidal
| C<sub>6v</sub>	
| 6mm
| *66
| [6]
| [[Polar point group|polar]]
| 12
| [[Dihedral group|dihedral]] <math>\mathbb{D}_{12} = \mathbb{Z}_6\rtimes\mathbb{Z}_2</math>
|-
| ditrigonal-dipyramidal
| D<sub>3h</sub>	
| {{overline|6}}m2 or {{overline|6}}2m
| *322
| [2,3]
| [[non-centrosymmetric]]
| 12
| [[Dihedral group|dihedral]] <math>\mathbb{D}_{12} = \mathbb{Z}_6\rtimes\mathbb{Z}_2</math>
|-
| dihexagonal-dipyramidal
| D<sub>6h</sub>	
| 6/mmm
| *622
| [2,6]
| [[centrosymmetric]]
| 24
| <math>\mathbb{D}_{12}\times\mathbb{Z}_2</math> 
|-
! rowspan=5 colspan=2 | [[cubic crystal system|cubic]]
| tetartoidal
| T
| 23
| 332
| [3,3]<sup>+</sup>
| [[Chirality (chemistry)|enantiomorphic]]
| 12
| [[alternating group|alternating]] <math>\mathbb{A}_4</math>
|-
| diploidal
| T<sub>h</sub>	
| m{{overline|3}}
| 3*2
| [3<sup>+</sup>,4]
| [[centrosymmetric]]
| 24
| <math>\mathbb{A}_4\times\mathbb{Z}_2</math> 
|-
| gyroidal
| O
| 432
| 432
| [4,3]<sup>+</sup>
| [[Chirality (chemistry)|enantiomorphic]]
| 24
| [[symmetric group|symmetric]] <math>\mathbb{S}_4</math>
|-
| [[tetrakis hexahedron|hextetrahedral]]
| T<sub>d</sub>	
| {{overline|4}}3m
| *332
| [3,3]
| [[non-centrosymmetric]]
| 24
| [[symmetric group|symmetric]] <math>\mathbb{S}_4</math>
|-
| [[disdyakis dodecahedron|hexoctahedral]]
| O<sub>h</sub>	
| m{{overline|3}}m
| *432
| [4,3]
| [[centrosymmetric]]
| 48
| <math>\mathbb{S}_4\times\mathbb{Z}_2</math> 
|}

The point symmetry of a structure can be further described as follows. Consider the points that make up the structure, and reflect them all through a single point, so that (''x'',''y'',''z'') becomes (−''x'',−''y'',−''z''). This is the 'inverted structure'. If the original structure and inverted structure are identical, then the structure is ''centrosymmetric''. Otherwise it is ''non-centrosymmetric''. Still, even in the non-centrosymmetric case, the inverted structure can in some cases be rotated to align with the original structure. This is a non-centrosymmetric ''achiral'' structure. If the inverted structure cannot be rotated to align with the original structure, then the structure is ''[[Chirality|chiral]]'' or ''enantiomorphic'' and its symmetry group is ''enantiomorphic''.<ref>{{cite journal|last=Flack|first=Howard D.|year=2003|title=Chiral and Achiral Crystal Structures|journal=Helvetica Chimica Acta|volume=86|issue=4|pages=905–921|citeseerx=10.1.1.537.266|doi=10.1002/hlca.200390109}}</ref>

A direction (meaning a line without an arrow) is called ''polar'' if its two-directional senses are geometrically or physically different. A symmetry direction of a crystal that is polar is called a ''polar axis''.{{Sfn|Hahn|2002|p=804}} Groups containing a polar axis are called ''[[polar point group|polar]]''. A polar crystal possesses a unique polar axis (more precisely, all polar axes are parallel). Some geometrical or physical property is different at the two ends of this axis: for example, there might develop a [[Polarization density|dielectric polarization]] as in [[Pyroelectricity|pyroelectric crystals]]. A polar axis can occur only in non-centrosymmetric structures. There cannot be a mirror plane or twofold axis perpendicular to the polar axis, because they would make the two directions of the axis equivalent.

The [[crystal structure]]s of chiral biological molecules (such as [[protein]] structures) can only occur in the 65 [[Chirality (chemistry)|enantiomorphic]] space groups (biological molecules are usually [[Chirality (chemistry)|chiral]]). 
<!--The protein assemblies themselves may have symmetries other than those given above because they are not intrinsically restricted by the [[Crystallographic restriction theorem]]. For example, the [[Rad52]] DNA binding protein has an 11-fold rotational symmetry (in human), however, it must form crystals in one of the 65 [[Chirality (chemistry)|enantiomorphic]] space groups given above. -->