40 



B' 



T80 



Aiao 



180 



t80 ' 



180" 



C 



Fig. 33. 



r\ 



mopolar axis ON, with a period , - - n being 2 or greater than 



n 



2, and by n binary axes situated in a plane perpendicular to ON 



7T 



and intersecting at angles of . 



These binary axes are homopolar, 

 but belong alternately to two diffe- 

 rent sets if n is an even number ; 

 and the axes are equivalent but 

 heteropolar if n is an odd num- 

 ber. The corresponding groups are 

 named dihedron-groups, and they 

 will in future generally be denoted 

 by the symbol D n . 



5. With respect to these 

 dihedron-groups D n , it will be 

 remembered that n can also have 

 the value 2. In this special case we have to deal with figures 

 which have three binary axes of three different kinds, and which 

 are all perpendicular to each other. Figure 33 

 will make this clear; obviously every-one 

 of the three axes will coincide only with 

 itself if the symme- 

 trical figure be sub- 

 jected to its charac- 

 teristic motions. 



In fig. 34 and 35 

 two polyhedra with 

 the symmetry of the 

 groups > 3 and Z) 6 

 respectively, are re- 

 produced as illustra- 

 tions of figures of 

 thiskind.The binary 

 axes are indicated, p . 



and it is easily Seen Hexagonal trapezohedron. 

 Trigonal trapezohedron. from ^ ^ an d jj-, 



that in the case of D 3 both ends of these binary axes are 

 non-equivalent, while in the case of D 6 they are equivalent, but 

 three of them have a function different from the three alter- 



