1.34: 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 n , of which lattice and translation vectors are elements. It is a 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 n is attached
(ii) For each point P of E n and for each vector r of V n there is exactly one point Q of E n for which PQ = r holds
(iii) If R is a third point of the point space, PQ + QR = PR