# Direct space

The direct space (or *crystal space*) is the *point space*, *E ^{n}*, in which the structures of finite real crystals are idealized as infinite perfect three-dimensional structures. To this space one associates the

*vector space*,

*V*, of which lattice and translation vectors are elements. It is a

^{n}*Euclidean*space where the scalar product of two vectors is defined. The two spaces are connected through the following relations:

(i) To any two points *P* and *Q* of the point space *E ^{n}* a vector

**PQ**=

**r**of the vector space

*V*is attached

^{n}(ii) For each point *P* of *E ^{n}* and for each vector

**r**of

*V*there is exactly one point

^{n}*Q*of

*E*for which

^{n}**PQ**=

**r**holds

(iii) If *R* is a third point of the point space, **PQ** + **QR** = **PR**

### See also

Section 8.1 of *International Tables of Crystallography, Volume A*