Fixed Points of Isometric Transformations

1 Fixed Points of Isometric Transformations. In many SRMs special values of the internal coordinates occur, which define NC X, Zk, Mk of higher covering symmetry than ( ) if F is such a point in the parameter space, then 

The relation of such special points to internal isometric transformations may be derived as follows. 

The last Eq. (2.66) implies that the mapping on the right hand side generates merely a permutation of the coordinate vectors, i.e. F (F) must be a covering operation of the NC (Xk( F), Zk, Mk. One therefore may state that every fixed point is connected to a covering symmetry operation not contained in ( ). If F E JF ( ) is an isometric transformation with fixed point, then also the period of F 

It should be pointed out that the fixed points play an important role in geometrical application of isometric groups, e.g. stereochemistry, cf. Sects. 3.4 and 3.5. 

---