In crystallography, a crystal system or crystal family or lattice system is one of several classes of space groups, lattices, point groups, or crystals. Informally, two crystals tend to be in the same crystal system if they have similar symmetries, though there are many exceptions to this.
Crystal systems, crystal families, and lattice systems are similar but slightly different, and there is widespread confusion between them: in particular the trigonal crystal system is often confused with the rhombohedral lattice system, and the term "crystal system" is sometimes used to mean "lattice system" or "crystal family".
Space groups and crystals are divided into 7 crystal systems according to their point groups, and into 7 lattice systems according to their Bravais lattices. Five of the crystal systems are essentially the same as five of the lattice systems, but the hexagonal and trigonal crystal systems differ from the hexagonal and rhombohedral lattice systems. The six crystal families are formed by combining the hexagonal and trigonal crystal systems into one hexagonal family, in order to eliminate this confusion.
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A lattice system is a class of lattices with the same point group. In three dimensions there are 7 lattice systems: triclinic, monoclinic, orthorhombic, tetragonal, rhombohedral, hexagonal, and cubic. The lattice system of a crystal or space group is determined by its lattice but not always by its point group.
A crystal system is a class of point groups. Two point groups are placed in the same crystal system if the sets of possible lattice systems of their space groups are the same. For many point groups there is only one possible lattice system, and in these cases the crystal system corresponds to a lattice system and is given the same name. However for the 5 point groups in the trigonal crystal class there are two possible lattice systems for their point groups: rhombohedral or hexagonal. In three dimensions there are 7 crystal systems: triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal, and cubic. The crystal system of a crystal or space group is determined by its point group but not always by its lattice.
A crystal family also consists of point groups and is formed by combining crystal systems whenever two crystal systems have space groups with the same lattice. In three dimensions a crystal family is almost the same as a crystal system (or lattice system), except that the hexagonal and trigonal crystal systems are combined into one hexagonal family. In three dimensions there are 6 crystal families: triclinic, monoclinic, orthorhombic, tetragonal, hexagonal, and cubic. The crystal family of a crystal or space group is determined by either its point group or its lattice, and crystal families are the smallest collections of point groups with this property.
In dimensions less than three there is no essential difference between crystal systems, crystal families, and lattice systems. There are 1 in dimension 0, 1 in dimension 1, and 4 in dimension 2, called oblique, rectangular, square, and hexagonal.
The relation between threedimensional crystal families, crystal systems, and lattice systems is shown in the following table:
Crystal family  Crystal system  Required symmetries of point group  point groups  space groups  bravais lattices  Lattice system 

Triclinic  None  2  2  1  Triclinic  
Monoclinic  1 twofold axis of rotation or 1 mirror plane  3  13  2  Monoclinic  
Orthorhombic  3 twofold axes of rotation or 1 twofold axis of rotation and two mirror planes.  3  59  4  Orthorhombic  
Tetragonal  1 fourfold axis of rotation  7  68  2  Tetragonal  
Hexagonal  Trigonal  1 threefold axis of rotation  5  7  1  Rhombohedral 
18  1  Hexagonal  
Hexagonal  1 sixfold axis of rotation  7  27  
Cubic  4 threefold axes of rotation  5  36  3  Cubic  
Total: 6  7  32  230  14  7 
The distribution of the 32 point groups into the 7 crystal systems is given in the following table.
crystal family  crystal system  point group / crystal class  Schönflies  HermannMauguin  orbifold  Type  order  structure 

triclinic  triclinicpedial  C_{1}  1  11  enantiomorphic polar  1  trivial  
triclinicpinacoidal  C_{i}  1  1x  centrosymmetric  2  cyclic  
monoclinic  monoclinicsphenoidal  C_{2}  2  22  enantiomorphic polar  2  cyclic  
monoclinicdomatic  C_{s}  m  1*  polar  2  cyclic  
monoclinicprismatic  C_{2h}  2/m  2*  centrosymmetric  4  2×cyclic  
orthorhombic  orthorhombicsphenoidal  D_{2}  222  222  enantiomorphic  4  dihedral  
orthorhombicpyramidal  C_{2v}  mm2  *22  polar  4  dihedral  
orthorhombicbipyramidal  D_{2h}  mmm  *222  centrosymmetric  8  2×dihedral  
tetragonal  tetragonalpyramidal  C_{4}  4  44  enantiomorphic polar  4  Cyclic  
tetragonaldisphenoidal  S_{4}  4  2x  4  cyclic  
tetragonaldipyramidal  C_{4h}  4/m  4*  centrosymmetric  8  2×cyclic  
tetragonaltrapezoidal  D_{4}  422  422  enantiomorphic  8  dihedral  
ditetragonalpyramidal  C_{4v}  4mm  *44  polar  8  dihedral  
tetragonalscalenoidal  D_{2d}  42m or 4m2  2*2  8  dihedral  
ditetragonaldipyramidal  D_{4h}  4/mmm  *422  centrosymmetric  16  2×dihedral  
hexagonal  trigonal  trigonalpyramidal  C_{3}  3  33  enantiomorphic polar  3  cyclic 
rhombohedral  S_{6} (C_{3i})  3  3x  centrosymmetric  6  cyclic  
trigonaltrapezoidal  D_{3}  32 or 321 or 312  322  enantiomorphic  6  dihedral  
ditrigonalpyramidal  C_{3v}  3m or 3m1 or 31m  *33  polar  6  dihedral  
ditrigonalscalahedral  D_{3d}  3m or 3m1 or 31m  2*3  centrosymmetric  12  dihedral  
hexagonal  hexagonalpyramidal  C_{6}  6  66  enantiomorphic polar  6  cyclic  
trigonaldipyramidal  C_{3h}  6  3*  6  cyclic  
hexagonaldipyramidal  C_{6h}  6/m  6*  centrosymmetric  12  2×cyclic  
hexagonaltrapezoidal  D_{6}  622  622  enantiomorphic  12  dihedral  
dihexagonalpyramidal  C_{6v}  6mm  *66  polar  12  dihedral  
ditrigonaldipyramidal  D_{3h}  6m2 or 62m  *322  12  dihedral  
dihexagonaldipyramidal  D_{6h}  6/mmm  *622  centrosymmetric  24  2×dihedral  
cubic  tetrahedral  T  23  332  enantiomorphic  12  Alternating  
diploidal  T_{h}  m3  3*2  centrosymmetric  24  2×alternating  
gyroidal  O  432  432  enantiomorphic  24  symmetric  
tetrahedral  T_{d}  43m  *332  24  symmetric  
hexoctahedral  O_{h}  m3m  *432  centrosymmetric  48  2×symmetric 
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 11fold rotational symmetry (in human), however, it must form crystals in one of the 11 enantiomorphic point groups given above.
The distribution of the 14 Bravais lattice types into 7 lattice systems is given in the following table.
The 7 lattice systems  The 14 Bravais Lattices  
triclinic (parallelepiped)  
monoclinic (right prism with parallelogram base; here seen from above)  simple  centered  
orthorhombic (cuboid)  simple  basecentered  bodycentered  facecentered 
tetragonal (square cuboid)  simple  bodycentered  
rhombohedral (trigonal trapezohedron) 
α=β=γ  
hexagonal (centered regular hexagon)  
cubic (isometric; cube) 
simple  bodycentered  facecentered  
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
where n_{1}, n_{2}, and n_{3} are integers and a_{1}, a_{2}, and a_{3} are three noncoplanar 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 lattice system only. They represent the maximum symmetry a structure with the translational symmetry concerned can have.
All crystalline materials must, by definition fit in one of these arrangements (not including quasicrystals).
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 (18011869), in 1842, who found that there were 15 Bravais lattices. This was corrected to 14 by A. Bravais in 1848.

