15 



, will however have not only "characteristic reflections" 

 it these will be necessarily accompanied by some "characteristic 

 >tion>" also. 



Tims in general we can maintain that the symmetry of a stereometrical 

 in- is known exactly, when the whole complex of its non-equivalent 

 inicteristic motions and reflections is determined. 

 3. More detailed investigation shows that reflection in a 



Fig. 8. 



lirror-plane is not the only way, in which a figure F can be trans- 

 formed into its mirror-image. In fig. 8 A BCD is an arbitrary, irregular 

 trahedron. If now a point P in space be chosen, and straight lines 



drawn through P from 

 ;very point of A BCD, and 

 :h respectively continued 

 an equal length beyond P, 

 number of points will be 

 ind, which joined together 

 a similar tetrahedron 

 'B'C'D'. This tetrahedron 



wever is not congruent 

 th A BCD , but is its mirror- 

 image; we say that it is ob- 

 tained from A BCD by inver- 

 sion with respect to the point 

 P, this point being called the 



centre of inversion. However it can be easily proved, that the transi- 

 tion of A BCD to A' B'C'D' could have taken place also in the follow- 

 ing way: suppose A BCD to be first rotated through an angle of 1'80 

 round an arbitrarily chosen axis LL', passing through P, and then the 

 tetrahedron in this new position to be reflected in a plane perpen- 



