40 



but of course an infinite number of types of these polyhedra are 

 possible, n having occasionally all 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. 34 and 35 some ima- 

 ginary "fruits" have been reproduced, in 

 order to make clear what they would look 

 Fig. 35. like; the symmetry of them is D z and Z) 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 hemimorphic 



Fig. 34- 



Fig. 36. 



Fig. 37. 



Fig. 38. 



development of forms. Therefore this hemimorphic form is no longer 

 observable here. In fig. 36, 37, and 

 38, three sections perpendicular to 

 the principal axis of such fruits, 

 having the symmetry D 3 , D 5 and 

 Z) 6 are drawn : here also the diffe- 

 rence 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 pro- 

 pellers, such as are used in aeropla- 

 nes, steamers, and in the laboratory 

 as apparatus for the stirring of 

 liquids. In fig. 39 such a propeller, used as a stirrer in a thermostate, 

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



Fig. 39- 

 Propeller. 



