Crystal system
In crystallography (the study of crystals), a crystal system is a set of point groups (a group of geometric symmetries with at least one fixed point). 32 point groups exist in three dimensions. Their corresponding space groups are assigned to a lattice system. Crystal systems that have space groups assigned to a common lattice system are combined into a crystal family. The seven crystal systems are triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal, and cubic. Informally, two crystals are in the same crystal system if they have similar symmetries (albeit many exceptions). Classifications. Crystals can be classified in other  ways, including crystal families and lattice systems. These make use of space groups, lattices, and point groups. The various classifications are often confused: in particular the trigonal crystal system is often confused with the rhombohedral lattice system, and the term "crystal system" is sometimes used to mean one of the others. Spaces with fewer than three dimensions have the same number of crystal systems, crystal families, and lattice systems. In one-dimensional space, there is one crystal system. In 2D space, there are four crystal systems: oblique, rectangular, square, and hexagonal. Crystal family. A crystal family is determined by lattices  and point groups. It is formed by combining crystal systems that have space groups assigned to a common lattice system. In three dimensions, the crystal families and systems are identical, except the hexagonal and trigonal crystal systems, which are combined into one hexagonal crystal family. The six crystal families are triclinic, monoclinic, orthorhombic, tetragonal, hexagonal, and cubic. Lattice system. A lattice system is a group of lattices with the same set of lattice point groups, which are subgroups of the arithmetic crystal classes. Space groups and crystals are classified as lattice systems according to their Bravais lattices. The 14 Bravais  lattices are grouped into seven lattice systems: triclinic, monoclinic, orthorhombic, tetragonal, rhombohedral, hexagonal, and cubic. Comparison. Five of the crystal systems are essentially the same as five of the lattice systems, while the hexagonal and trigonal crystal systems differ. Most point groups are assigned to a single lattice system, in which case both the crystal and lattice systems have the same name. However, five point groups are assigned to two lattice systems, rhombohedral and hexagonal, because both exhibit threefold rotational symmetry. These point groups are assigned to the trigonal crystal system. The relation between three-dimensional crystal families, crystal systems  and lattice systems is shown in the following table: Crystal classes. The 7 crystal systems consist of 32 crystal classes (corresponding to the 32 crystallographic point groups) as shown in the following table below: 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 "chiral" or "enantiomorphic" and its symmetry group is "enantiomorphic". 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". Groups containing a polar axis are called "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 dielectric polarization as in 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 structures of chiral biological molecules (such as protein structures) can only occur in the 65 enantiomorphic space groups (biological molecules are usually chiral). Bravais lattices. There are seven different kinds of  crystal systems, and each kind of crystal 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 lattice systems and crystal families is given in the following table. In geometry and crystallography, a Bravais lattice is a category of translative symmetry groups (also known as lattices) in three directions. Such symmetry groups  consist of translations by vectors of the form where "n", "n", and "n" are integers and a, a, and a 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 represent the maximum symmetry a structure with the given translational symmetry can have. All crystalline materials (not including quasicrystals) 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 A. Bravais in 1848. In four-dimensional space. ‌The four-dimensional unit cell is defined by four edge lengths ("a", "b", "c", "d") and six interaxial angles ("α", "β", "γ", "δ", "ε", "ζ"). The following conditions for the lattice  parameters define 23 crystal families The names here are given according to Whittaker. They are almost the same as in Brown "et al", with exception for names of the crystal families 9, 13, and 22. The names for these three families according to Brown "et al" are given in parenthesis. The relation between four-dimensional crystal families, crystal systems, and lattice systems is shown in the following table. Enantiomorphic systems are marked with an asterisk. The number of enantiomorphic pairs is given in parentheses. Here the term "enantiomorphic" has a different meaning than in the table for three-dimensional crystal  classes. The latter means, that enantiomorphic point groups describe chiral (enantiomorphic) structures. In the current table, "enantiomorphic" means that a group itself (considered as a geometric object) is enantiomorphic, like enantiomorphic pairs of three-dimensional space groups P3 and P3, P422 and P422. Starting from four-dimensional space, point groups also can be enantiomorphic in this sense.