A crystal system is a category of space groups, which characterize symmetry of structures in three dimensions with translational symmetry in three directions, having a discrete symmetry group. A major application is in crystallography, to categorize crystals, but by itself the topic is one of 3D Euclidean geometry.

There are 7 crystal systems:

• Triclinic, all cases not satisfying the requirements of any other system; thus there is no other symmetry than translational symmetry, or the only extra kind is inversion.
• Monoclinic, requires either 1 two-fold axis of rotation or 1 mirror plane.
• Orthorhombic, requires either 3 two-fold axes of rotation or 1 two fold axis of rotation and two mirror planes.
• Tetragonal, requires 1 four-fold axis of rotation.
• Rhombohedral, also called trigonal, requires 1 three-fold axis of rotation.
• Hexagonal, requires 1 six-fold axis of rotation.
• Isometric or cubic, requires 4 three-fold axes of rotation.

There are 2, 13, 59, 68, 25, 27, and 36 space groups per crystal system, respectively, together 230. The following mini-table gives a breakdown of the various different things per crystal system;

Crystal system	'''No. of point groups	No. of bravais lattices	No. of space groups
Triclinic	2	1	2
Monoclinic	3	2	13
Orthorhombic	3	4	59
Tetragonal	7	2	68
Rhombohedral(Trigonal)	5	1	25
Hexagonal	7	1	27
Cubic	5	3	36
Total	32	14	230

Within a crystal system there are two ways of categorizing space groups:

• by the linear parts of symmetries, i.e. by crystal class, also called crystallographic point group; each of the 32 crystal classes applies for one of the 7 crystal systems
• by the symmetries in the translation lattice, i.e. by Bravais lattice; each of the 14 Bravais lattices applies for one of the 7 crystal systems.

The 73 symmorphic space groups (see space group) are largely combinations, within each crystal system, of each applicable point group with each applicable Bravais lattice: there are 2, 6, 12, 14, 5, 7, and 15 combinations, respectively, together 61.

Crystallographic point group

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A symmetry group consists of isometric affine transformations; each is given by an orthogonal matrix and a translation vector (which may be the zero vector). Space groups can be grouped by the matrices involved, i.e. ignoring the translation vectors (see also Euclidean group). This corresponds to discrete symmetry groups with a fixed point. There are infinitely many of these point groups in three dimensions. However, only part of these are compatible with translational symmetry: the crystallographic point groups. This is expressed in the crystallographic restriction theorem. (In spite of these names, this is a geometric limitation, not just a physical one.)

The point group of a crystal, among other things, determines the symmetry of the crystal's optical properties. For instance, one knows whether it is birefringent, or whether it shows the Pockels effect, simply by knowing its point group.

Overview of point groups by crystal system

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-	crystal system	point group / crystal class	Schönflies	Hermann-Mauguin	orbifold	Type	-	triclinic	triclinic-pedial	C1	$1\backslash$	11	enantiomorphic polar	-	triclinic-pinacoidal	Ci	$\backslash bar\{1\}$	1x	centrosymmetric	-	monoclinic	monoclinic-sphenoidal	C2	$2\backslash$	22	enantiomorphic polar	-	monoclinic-domatic	Cs	$m\backslash$	1*	polar	-	monoclinic-prismatic	C2h	$2/m\backslash$	2*	centrosymmetric	-	orthorhombic	orthorhombic-sphenoidal	D2	$222\backslash$	222	enantiomorphic	-	orthorhombic-pyramidal	C2v	$mm2\backslash$	*22	polar	-	orthorhombic-bipyramidal	D2h	$mmm\backslash$	*222	centrosymmetric	-	tetragonal	tetragonal-pyramidal	C4	$4\backslash$	44	enantiomorphic polar	-	tetragonal-disphenoidial	S4	$\backslash bar\{4\}$	2x		-	tetragonal-dipyramidal	C4h	$4/m\backslash$	4*	centrosymmetric	-	tetragonal-trapezoidal	D4	$422\backslash$	422	enantiomorphic	-	ditetragonal-pyramidal	C4v	$4mm\backslash$	*44	polar	-	tetragonal-scalenoidal	D2d	$\backslash bar\{4\}2m\backslash$ or $\backslash bar\{4\}m2$	2*2		-	ditetragonal-dipyramidal	D4h	$4/mmm\backslash$	*422	centrosymmetric	-	rhombohedral (trigonal)	trigonal-pyramidal	C3	$3\; \backslash !$	33	enantiomorphic polar	-	rhombohedral	S6 (C3i)	$\backslash bar\{3\}$	3x	centrosymmetric	-	trigonal-trapezoidal	D3	$32\backslash$ or $321\backslash$ or $312\backslash$	322	enantiomorphic	-	ditrigonal-pyramidal	C3v	$3m\backslash$or $3m1\backslash$ or $31m\backslash$	*33	polar	-	ditrigonal-scalahedral	D3d	$\backslash bar\{3\}\; m\backslash$ or $\backslash bar\{3\}\; m\; 1$ or $\backslash bar\{3\}\; 1\; m$	2*3	centrosymmetric	-	hexagonal	hexagonal-pyramidal	C6	$6\backslash$	66	enantiomorphic polar	-	trigonal-dipyramidal	C3h	$\backslash bar\{6\}$	3*		-	hexagonal-dipyramidal	C6h	$6/m\backslash$	6*	centrosymmetric	-	hexagonal-trapezoidal	D6	$622\backslash$	622	enantiomorphic	-	dihexagonal-pyramidal	C6v	$6mm\backslash$	*66	polar	-	ditrigonal-dipyramidal	D3h	$\backslash bar\{6\}m2$ or $\backslash bar\{6\}2m$	*322		-	dihexagonal-dipyramidal	D6h	$6/mmm\backslash$	*622	centrosymmetric	-	cubic	tetartoidal	T	$23\backslash$	332	enantiomorphic	-	diploidal	Th	$m\backslash bar\{3\}\backslash$	3*2	centrosymmetric	-	gyroidal	O	$432\backslash$	432	enantiomorphic	-	tetrahedral	Td	$\backslash bar\{4\}3m$	*332		-	hexoctahedral	Oh	$m\backslash bar\{3\}m$	*432	centrosymmetric

The crystal structures of biological molecules (such as protein structures) can only occur in the 11 enantiomorphic point groups, as biological molecules are invariably 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 11 enantiomorphic point groups given above.

Classification of lattices

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Crystal system	Lattices
triclinic (parallelepiped)
monoclinic (right prism with parallelogram base; here seen from above)	simple	centered
orthorhombic (cuboid)	simple	base-centered	body-centered	face-centered
tetragonal (square cuboid)	simple	body-centered
rhombohedral (trigonal) (3-sided trapezohedron)
hexagonal (centered regular hexagon)
cubic (isometric; cube)	simple	body-centered	face-centered

In geometry and crystallography, a Bravais lattice is a category of symmetry groups for translational symmetry in three directions, or correspondingly, a category of translation lattices.

Such symmetry groups consist of translations by vectors of the form

$\backslash mathbf\{R\}\; =\; n\_1\; \backslash mathbf\{a\}\_1\; +\; n\_2\; \backslash mathbf\{a\}\_2\; +\; n\_3\; \backslash mathbf\{a\}\_3,$

where n₁, n₂, and n₃ are integers and a₁, a₂, and a₃ are three non-coplanar vectors, called primitive vectors.

These lattices are classified by space group of the translation lattice itself; there are 14 Bravais lattices in three dimensions; each can apply in one crystal system only. They represent the maximum symmetry a structure with the translational symmetry concerned can have.

All crystalline materials recognised till now (not including quasicrystals) fit in 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 (1801-1869), in 1842, who found that there were 15 Bravais lattices. This was corrected to 14 by A. Bravais in 1848.

See also

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• Crystal structure
• Point group
• Système_cristallin#Les_230_groupes_d.27espace (in French)

External links

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• Overview of the 32 groups
• Ditto - uses a different notation
• Mineral galleries - Symmetry

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