39 



180" 



Aiao' 



180' 



c 



Fig. 31. 



5. With respect to these dihodron-groups D n , it will l>- i nu -in- 

 red that n can also have the value 2. In this special case we 

 ve to deal with figures 

 ,-hich have three binary axes of C 



ree different kinds, and which 

 all perpendicular to each 

 icr. Figure jl will make this 

 ; obviously every-one of the teo' A 



axes will coincide only with 

 jlf if the symmetrical figure 

 subjected to its characteristic 

 lotions. 



In fig. 32 and 33 two poly- 



jdra with the symmetry of the 



roups Z> 3 and Z? 6 respectively, 



re reproduced as illustrations 



)f figures of this kind. The 



try axes are indicated, and it is easily seen from fig 32 and 33, 

 lat in the case of Z? 3 both ends of these binary axes are in fact non- 

 equivalent, while in 

 the case of D^ they 

 are equivalent, but 

 three of them have 

 a function different 

 from the three alter- 

 nating ones. More- 

 over to every face 

 Z there corresponds 

 an equivalent face 

 Z' , as a consequence 

 of the existence of 

 the binary axis 

 both faces forming 

 together a dihedron. 

 It is because of this peculiarity 

 polyhedra of this kind, that the 



Fi S- 33- 



Iexagonal 



of the 

 groups 



Trigonal trapezohedron. themselves got the name of dihedron-groups. 

 Polyhedra of this symmetry are limited by irregular four-sided faces, 

 and they are therefore commonly called trapezohedra. The figures 

 32 and 33 are instances of such trigonal and hexagonal trapezohedra; 



