Editing Crystal structure (section)

===Crystal systems===
{{see also|Crystallographic point group#Isomorphisms}}

A crystal system is a set of point groups in which the point groups themselves and their corresponding space groups are assigned to a lattice system. Of the 32 point groups that exist in three dimensions, most are assigned to only one lattice system, in which case the crystal system and lattice system both have the same name. However, five point groups are assigned to two lattice systems, rhombohedral and hexagonal, because both lattice systems exhibit threefold rotational symmetry. These point groups are assigned to the trigonal crystal system.

{| class=wikitable
|+Overview of crystal systems
|-
!scope="col"| Crystal family
!scope="col"| Crystal system
!scope="col"| [[Point group]] / Crystal class
!scope="col"| [[Schönflies notation|Schönflies]]
!scope="col"| Point symmetry
!scope="col"| [[Symmetry number|Order]]
!scope="col"| [[Group theory#Abstract groups|Abstract group]]
|-
!scope="row" rowspan=2 colspan=2| [[triclinic crystal system|triclinic]]
| pedial	
| C{{sub|1}}
| [[Chirality (chemistry)|enantiomorphic]] [[Polar point group|polar]]
| 1
| trivial <math>\mathbb{Z}_1</math>
|-
| pinacoidal
| C{{sub|i}} (S{{sub|2}})
| [[centrosymmetric]]
| 2
| [[Cyclic group|cyclic]] <math>\mathbb{Z}_2</math>
|-
!scope="row" rowspan=3 colspan=2 | [[monoclinic crystal system|monoclinic]]
| sphenoidal
| C{{sub|2}}
| [[Chirality (chemistry)|enantiomorphic]] [[Polar point group|polar]]
| 2
| [[Cyclic group|cyclic]] <math>\mathbb{Z}_2</math>
|-
| domatic
| C{{sub|s}} (C{{sub|1h}})
| [[Polar point group|polar]]
| 2
| [[Cyclic group|cyclic]] <math>\mathbb{Z}_2</math>
|-
| [[prism (geometry)|prismatic]]
| C{{sub|2h}}	
| [[centrosymmetric]]
| 4
| [[Klein four-group|Klein four]] <math>\mathbb{V} = \mathbb{Z}_2\times\mathbb{Z}_2</math>
|-
!scope="row" rowspan=3 colspan=2| [[orthorhombic crystal system|orthorhombic]]
| rhombic-disphenoidal
| D{{sub|2}} (V)
| [[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}}	
| [[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}} (V{{sub|h}})	
| [[centrosymmetric]]
| 8
| <math>\mathbb{V}\times\mathbb{Z}_2</math>
|-
!scope="row" rowspan=7 colspan=2| [[tetragonal crystal system|tetragonal]]
| tetragonal-pyramidal	
| C{{sub|4}}
| [[Chirality (chemistry)|enantiomorphic]] [[Polar point group|polar]]
| 4
| [[Cyclic group|cyclic]] <math>\mathbb{Z}_4</math>
|-
| tetragonal-disphenoidal
| S{{sub|4}}	
| [[non-centrosymmetric]]
| 4
| [[Cyclic group|cyclic]] <math>\mathbb{Z}_4</math>
|-
| tetragonal-dipyramidal
| C{{sub|4h}}	
| [[centrosymmetric]]
| 8
| <math>\mathbb{Z}_4\times\mathbb{Z}_2</math>
|-
| tetragonal-trapezohedral
| D{{sub|4}}	
| [[Chirality (chemistry)|enantiomorphic]]
| 8
| [[Dihedral group|dihedral]] <math>\mathbb{D}_8 = \mathbb{Z}_4\rtimes\mathbb{Z}_2</math>
|-
| ditetragonal-pyramidal
| C{{sub|4v}}	
| [[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}} (V{{sub|d}})
| [[non-centrosymmetric]]
| 8
| [[Dihedral group|dihedral]] <math>\mathbb{D}_8 = \mathbb{Z}_4\rtimes\mathbb{Z}_2</math>
|-
| ditetragonal-dipyramidal
| D{{sub|4h}}	
| [[centrosymmetric]]
| 16
| <math>\mathbb{D}_8\times\mathbb{Z}_2</math>
|-
!scope="row" rowspan=12|[[hexagonal crystal family|hexagonal]] !! rowspan=5 | trigonal 
| trigonal-pyramidal	
| C{{sub|3}}
| [[Chirality (chemistry)|enantiomorphic]] [[Polar point group|polar]]
| 3
| [[Cyclic group|cyclic]] <math>\mathbb{Z}_3</math>
|-
| rhombohedral
| C{{sub|3i}} (S{{sub|6}})
| [[centrosymmetric]]
| 6
| [[Cyclic group|cyclic]] <math>\mathbb{Z}_6 = \mathbb{Z}_3\times\mathbb{Z}_2</math>
|-
| trigonal-trapezohedral
| D{{sub|3}}	
| [[Chirality (chemistry)|enantiomorphic]]
| 6
| [[Dihedral group|dihedral]] <math>\mathbb{D}_6 = \mathbb{Z}_3\rtimes\mathbb{Z}_2</math>
|-
| ditrigonal-pyramidal
| C{{sub|3v}}	
| [[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}}	
| [[centrosymmetric]]
| 12
| [[Dihedral group|dihedral]] <math>\mathbb{D}_{12} = \mathbb{Z}_6\rtimes\mathbb{Z}_2</math>
|-
!scope="row" rowspan=7 | hexagonal
| hexagonal-pyramidal	
| C{{sub|6}}
| [[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}}
| [[non-centrosymmetric]]
| 6
| [[Cyclic group|cyclic]] <math>\mathbb{Z}_6 = \mathbb{Z}_3\times\mathbb{Z}_2</math>
|-
| hexagonal-dipyramidal
| C{{sub|6h}}	
| [[centrosymmetric]]
| 12
| <math>\mathbb{Z}_6\times\mathbb{Z}_2</math>
|-
| hexagonal-trapezohedral
| D{{sub|6}}	
| [[Chirality (chemistry)|enantiomorphic]]
| 12
| [[Dihedral group|dihedral]] <math>\mathbb{D}_{12} = \mathbb{Z}_6\rtimes\mathbb{Z}_2</math>
|-
| dihexagonal-pyramidal
| C{{sub|6v}}	
| [[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}}	
| [[non-centrosymmetric]]
| 12
| [[Dihedral group|dihedral]] <math>\mathbb{D}_{12} = \mathbb{Z}_6\rtimes\mathbb{Z}_2</math>
|-
| dihexagonal-dipyramidal
| D{{sub|6h}}	
| [[centrosymmetric]]
| 24
| <math>\mathbb{D}_{12}\times\mathbb{Z}_2</math> 
|-
!scope="row" rowspan=5 colspan=2 | [[cubic crystal system|cubic]]
| tetartoidal
| T 
| [[Chirality (chemistry)|enantiomorphic]]
| 12
| [[alternating group|alternating]] <math>\mathbb{A}_4</math>
|-
| diploidal
| T{{sub|h}}	
| [[centrosymmetric]]
| 24
| <math>\mathbb{A}_4\times\mathbb{Z}_2</math> 
|-
| gyroidal
| O
| [[Chirality (chemistry)|enantiomorphic]]
| 24
| [[symmetric group|symmetric]] <math>\mathbb{S}_4</math>
|-
| hextetrahedral
| T{{sub|d}}	
| [[non-centrosymmetric]]
| 24
| [[symmetric group|symmetric]] <math>\mathbb{S}_4</math>
|-
| hexoctahedral
| O{{sub|h}}	
| [[centrosymmetric]]
| 48
| <math>\mathbb{S}_4\times\mathbb{Z}_2</math> 
|}
{{clear}}
In total there are seven crystal systems: triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal, and cubic.

==== Point groups ====
The [[crystallographic point group]] or ''crystal class'' is the mathematical group comprising the symmetry operations that leave at least one point unmoved and that leave the appearance of the crystal structure unchanged. These symmetry operations include

*''Reflection'', which reflects the structure across a ''reflection plane''
*''Rotation'', which rotates the structure a specified portion of a circle about a ''rotation axis''
*''Inversion'', which changes the sign of the coordinate of each point with respect to a ''center of symmetry'' or ''inversion point''
*''[[Improper rotation]]'', which consists of a rotation about an axis followed by an inversion.

Rotation axes (proper and improper), reflection planes, and centers of symmetry are collectively called ''symmetry elements''. There are 32 possible crystal classes. Each one can be classified into one of the seven crystal systems.