International
Tables for Crystallography Volume B Reciprocal space Edited by U. Shmueli © International Union of Crystallography 2006 
International Tables for Crystallography (2006). Vol. B, ch. 1.3, pp. 4041

Let be the subset of consisting of those points with (signed) integer coordinates; it is an ndimensional lattice, i.e. a free Abelian group on n generators. A particularly simple set of n generators is given by the standard basis of , and hence will be called the standard lattice in . Any other `nonstandard' ndimensional lattice Λ in is the image of this standard lattice by a general linear transformation.
If we identify any two points in whose coordinates are congruent modulo , i.e. differ by a vector in , we obtain the standard ntorus . The latter may be viewed as , i.e. as the Cartesian product of n circles. The same identification may be carried out modulo a nonstandard lattice Λ, yielding a nonstandard ntorus . The correspondence to crystallographic terminology is that `standard' coordinates over the standard 3torus are called `fractional' coordinates over the unit cell; while Cartesian coordinates, e.g. in ångströms, constitute a set of nonstandard coordinates.
Finally, we will denote by I the unit cube and by the subset