text: In crystallography, a fractional coordinate system is used in order to better reflect the symmetry of the underlying lattice of a crystal pattern (or any other periodic pattern in space). In a fractional coordinate system the basis vectors of the coordinate system are chosen to be lattice vectors and the basis is then termed a crystallographic basis (or lattice basis). 
  In a lattice basis, any lattice vector  can be represented as, 
  There are an infinite number of lattice bases for a crystal pattern. However, these can be chosen in such a way that the simplest description of the pattern can be obtained. These bases are used in the International Tables of Crystallography Volume A and are termed conventional bases. A lattice basis  is called primitive if the basis vectors are lattice vectors and all lattice vectors  can be expressed as,
  However, the conventional basis for a crystal pattern is not always chosen to be primitive. Instead, it is chosen so the number of orthogonal basis vectors is maximized. This results in some of the coefficients of the equations above being fractional. A lattice in which the conventional basis is primitive is called a primitive lattice, while a lattice with a non-primitive conventional basis is called a centered lattice.
  The choice of an origin and a basis implies the choice of a unit cell which can further be used to describe a crystal pattern. The unit cell is defined as the parallelotope (i.e., generalization of a parallelogram (2D) or parallelepiped (3D) in higher dimensions) in which the coordinates of all points are such that, .
  Furthermore, points outside of the unit cell can be transformed inside of the unit cell through standardization, the addition or subtraction of integers to the coordinates of points to ensure . In a fractional coordinate system, the lengths of the basis vectors  and the angles between them  are called the lattice parameters (lattice constants) of the lattice. In two- and three-dimensions, these correspond to the lengths and angles between the edges of the unit cell.
  The fractional coordinates of a point in space  in terms of the lattice basis vectors is defined as,