From Online Dictionary of Crystallography
Réseau direct (Fr). Direktes Gitter (Ge). Red directa (Sp). Reticolo diretto (It). 直格子 (Ja)
The direct lattice represents the triple periodicity of the ideal infinite perfect periodic structure that can be associated to the structure of a finite real crystal. To express this periodicity one calls crystal pattern an object in point space En (direct space) that is invariant with respect to three linearly independent translations, t1, t2 and t3. One distinguishes two kinds of lattices, the vector lattices and the point lattices.
Any translation t = ui ti (ui arbitrary integers) is also a translation of the pattern and the infinite set of all translation vectors of a crystal pattern is the vector lattice L of this crystal pattern.
Given an arbitrary point P in point space, the set of all the points Pi deduced from one of them by a translation PPi = ti of the vector lattice L is called the point lattice.
A basis a, b, c of the vector space Vn is a crystallographic basis of the vector lattice L if every integral linear combination t = u a + v b + w c is a lattice vector of L. It is called a primitive basis if every lattice vector t of L may be obtained as an integral linear combination of the basis vectors, a, b, c. Referred to any crystallographic basis the coefficients of each lattice vector are either integral or rational, while in the case of a primitive basis they are integral. Non-primitive bases are used conventionally to describe centred lattices.
The parallelepiped built on the basis vectors is the unit cell. Its volume is given by the triple scalar product, V = (a, b, c).
If the basis is primitive, the unit cell is called the primitive cell. It contains only one lattice point. If the basis is non-primitive, the unit cell is a multiple cell and it contains more than one lattice point. The multiplicity of the cell is given by the ratio of its volume to the volume of a primitive cell.
The generalization of the notion of point and vector lattices to n-dimensional space is given in Section 8.1 of International Tables of Crystallography, Volume A