16 



mirror-image, will, however, have not only "characteristic reflec- 

 tions", but these will necessarily be accompanied by some "charac- 

 teristic motions" also. 



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

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

 characteristic motions and reflections is determined. 



3. More detailed investigation shows that reflection in a 



C' 



D 



Fig. 10. 



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

 formed into its mirror-image. In fig. 10 A BCD is an arbitrary irregular 

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



be drawn through P from 

 every point of A BCD, and 

 each respectively continued 

 to an equal length beyond P, 

 a number of points will be 

 found, which joined together 

 form a similar tetrahedron 

 A'B'C'D'. This tetrahedron, 

 however, is not congruent 

 with ABCD,but is its mirror- 

 image; we say that it is ob- 

 tained from A BCD by inver- 

 Fi g . 11. 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 ABCD to A'B'C'D' could have also taken place in the follow- 

 ing way: suppose ABCD to be first rotated through an angle of 180 

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

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



