text: In crystallography, a fractional coordinate system (crystal coordinate system) is a coordinate system in which basis vectors used to the describe the space are the lattice vectors of a crystal (periodic) pattern. The selection of an origin and a basis define a unit cell, a parallelotope (i.e., generalization of a parallelogram (2D) or parallelepiped (3D) in higher dimensions) defined by the lattice basis vectors  where  is the dimension of the space. These basis vectors are described by lattice parameters (lattice constants) consisting of the lengths of the lattice basis vectors   and the angles between them .  
  Most cases in crystallography involve two- or three-dimensional space.  In the three-dimensional case, the basis vectors  are commonly displayed as  with their lengths denoted by  respectively, and the angles denoted by , where conventionally,  is the angle between  and ,  is the angle between  and , and  is the angle between  and .