Editing Crystal system (section)

==Bravais lattices==
{{main|Bravais lattice}}
There are seven different kinds of lattice systems, and each kind of lattice system has four different kinds of centerings (primitive, base-centered, body-centered, face-centered). However, not all of the combinations are unique; some of the combinations are equivalent while other combinations are not possible due to symmetry reasons. This reduces the number of unique lattices to the 14 Bravais lattices.

The distribution of the 14 Bravais lattices into 7 lattice systems is given in the following table.

{| class=wikitable
! rowspan=2| Crystal family
! rowspan=2| Lattice system
! rowspan=2| Point group <br />([[Schoenflies notation|Schönflies notation]])
! colspan=4| 14 Bravais lattices
|-
! Primitive (P)
! Base-centered (S)
! Body-centered (I)
! Face-centered (F)
|- align=center
! colspan=2| [[Triclinic]] (a)
| C<sub>i</sub>
| [[File:Triclinic.svg|80px|Triclinic]]
aP
|
|
|
|- align=center
! colspan=2| [[Monoclinic]] (m)
| C<sub>2h</sub>
| [[File:Monoclinic.svg|80px|Monoclinic, simple]]
mP
| [[File:Base-centered monoclinic.svg|80px|Monoclinic, centered]]
mS
|
|
|- align=center
! colspan=2| [[Orthorhombic]] (o)
| D<sub>2h</sub>
| [[File:Orthorhombic.svg|80px|Orthorhombic, simple]]
oP
| [[File:Orthorhombic-base-centered.svg|80px|Orthorhombic, base-centered]]
oS
| [[File:Orthorhombic-body-centered.svg|80px|Orthorhombic, body-centered]]
oI
| [[File:Orthorhombic-face-centered.svg|80px|Orthorhombic, face-centered]]
oF
|- align=center
! colspan=2| [[Tetragonal]] (t)
| D<sub>4h</sub>
| [[File:Tetragonal.svg|80px|Tetragonal, simple]]
tP
|
| [[File:Tetragonal-body-centered.svg|80px|Tetragonal, body-centered]]
tI
|
|- align=center
! rowspan=2| [[Hexagonal crystal family|Hexagonal]] (h)
! Rhombohedral
| D<sub>3d</sub>
| [[File:Rhombohedral.svg|80px|Rhombohedral]]
hR
|
|
|
|- align=center
! Hexagonal
| D<sub>6h</sub>
| [[File:Hexagonal latticeFRONT.svg|80px|Hexagonal]]
hP
|
|
|
|- align=center
! colspan=2| [[Cubic crystal system|Cubic]] (c)
| O<sub>h</sub>
| [[File:Cubic.svg|80px|Cubic, simple]]
cP
|
| [[File:Cubic-body-centered.svg|80px|Cubic, body-centered]]
cI
| [[File:Cubic-face-centered.svg|80px|Cubic, face-centered]]
cF
|}
{{Clear}}

In [[geometry]] and [[crystallography]], a '''Bravais lattice''' is a category of [[translational symmetry|translative]] [[symmetry group]]s (also known as [[Lattice (group)|lattice]]s) in three directions.

Such symmetry groups consist of translations by vectors of the form

:'''R''' = ''n''<sub>1</sub>'''a'''<sub>1</sub> + ''n''<sub>2</sub>'''a'''<sub>2</sub> + ''n''<sub>3</sub>'''a'''<sub>3</sub>,

where ''n''<sub>1</sub>, ''n''<sub>2</sub>, and ''n''<sub>3</sub> are [[integer]]s and '''a'''<sub>1</sub>, '''a'''<sub>2</sub>, and '''a'''<sub>3</sub> are three non-coplanar vectors, called ''primitive vectors''.

These lattices are classified by the  [[space group]] of the lattice itself, viewed as a collection of points; there are 14 Bravais lattices in three dimensions; each belongs to one lattice system only. They{{clarify|date=January 2019}} represent the maximum symmetry a structure with the given translational symmetry can have.

All crystalline materials (not including [[quasicrystal]]s) must, by definition, fit into one of these arrangements.

For convenience a Bravais lattice is depicted by a unit cell which is a factor 1, 2, 3, or 4 larger than the [[primitive cell]]. Depending on the symmetry of a crystal or other pattern, the [[fundamental domain]] is again smaller, up to a factor 48.

The Bravais lattices were studied by [[Moritz Ludwig Frankenheim]] in 1842, who found that there were 15 Bravais lattices. This was corrected to 14 by [[Auguste Bravais|A. Bravais]] in 1848<!-- or 1849 or 1850, Britannica has two different years-->.