41 



nating ones. Moreover, 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 peculia- 

 rity of the polyhedra of this kind, that the groups themselves received 

 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 j^ and 75 are instances 

 of such trigonal and hexagonal trapezohedra; but of 

 course an infinite number of types of these 

 polyhedra are possible, n having any one 

 of the values from 2 to oo. 



In nature there will perhaps be objects 

 having the symmetry of the groups D n . 

 Up till now, however, the existence of no 

 example of this kind has been proved with 

 certainty; hence in fig. 36 and 77 some Flg- 

 'fruits" have been reproduced, in order to make clear 

 what they would look like ; the symmetry of them is D 3 and D 6 

 respectively. The principal difference from the case of the cyclic 

 groups consists in the fact that the principal axis ON is no longer 

 heteropolar, as was the case in the cyclic symmetry with its hemi- 



Fig. 36, 

 imaginary 



Fig. 38. 



Fig. 39. 



morphic development of forms. Therefore this hemimorphic form 

 is no longer observable here. In fig. 38, jp, and 40, three sections 

 perpendicular to the principal axis of such fruits, having the symme- 

 try Z) 3 , D 5 and Z) 6 are drawn : here also the difference in the function 

 of the binary axes for both cases, when n is odd or is even, is once 

 more clearly demonstrated. As instances of objects having the 

 symmetry D n , attention may be drawn to the propellers, such as 

 are used in aeroplanes, steamers, and in the laboratory as apparatus 

 for the stirring of liquids. In fig. 41 such a propeller, used as a stirrer 

 in a thermostate, is shown in elevation; its symmetry is evidently Z) 4 . 



