A perfect infinite crystal possesses a lattice, an infinite set of points generated by three non-parallel vectors, such that each point is identical in itself and its surroundings.

With each lattice point may be associated a number of atoms. If their co-ordinates relative to the lattice point are given, together with the lengths and directions of the lattice vectors chosen to define the axes of reference, the complete structure is defined.

The magnitudes a, b, c are called lattice parameters.

Position coordinates x, y, z are commonly expressed as fractions of the lattice parameters a, b, c; the parallelepiped defined by the lattice vectors a, b, c, is the unit cell.

Symmetry elements may be present imposing certain relations:

1. Between lattice parameters
2. And between position coordinates of different atoms

Axes of reference are generally chosen in accordance with the symmetry. As a result of (i), and with a conventional choice of axes, crystals are classified into systems as follows:

Where is the angle between b and c, and similarly for β and γ. Accidental equality of unit cell edges, and special values of inter axial angles not required by the symmetry, are disregarded in making this classification.