Crystal system

The seven crystal systems are triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal, and cubic.

However, five point groups are assigned to two lattice systems, rhombohedral and hexagonal, because both exhibit threefold rotational symmetry.

[1] A direction (meaning a line without an arrow) is called polar if its two-directional senses are geometrically or physically different.

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.

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).

aP mP mS oP oS oI oF tP tI hR hP cP cI cF In geometry and crystallography, a Bravais lattice is a category of translative symmetry groups (also known as lattices) in three directions.

All crystalline materials (not including quasicrystals) must, by definition, fit into one of these arrangements.

Depending on the symmetry of a crystal or other pattern, the fundamental domain is again smaller, up to a factor 48.

[3][4] ‌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.

Here the term "enantiomorphic" has a different meaning than in the table for three-dimensional crystal classes.

Starting from four-dimensional space, point groups also can be enantiomorphic in this sense.