54 



may also be represented as resulting from the existence of an axis of the 



n 

 first order A n with a period-number , combined with a symmetry -centre. 



~2 2 



c. If, however, n and are both even numbers, the axis A n cannot be 



replaced by any other symmetry-element, or by any combination of them. 

 As illustrations of figures and objects having the symmetry 

 of the groups C 6 , C 3 , and C 4 respectively, we give here in fig. 53, 

 54. and 55, the images of some polyhedra. The first represents the 

 crystalform of dioptase: CuH 2 SiO^, and it is at once seen that the 

 axis A 6 is, as an axis of the first order only a ternary one, while 



an inversion-centre is 



combined with it. 



Of the groups C 3 and C 4 



we can only give some 



imaginary forms, because 



no real representatives of 



those groups have been 



found in the world of 



crystalline matter up to, 



this date. But in any 



case it may be seen 



from these figures, that 



the symmetry of~C 3 is the same, as if an axis of 

 the first order A% were present with a reflecting 

 plane perpendicular to it. In the same way it will 

 be obvious that in fig. 55 the special sym- 

 metry of the polyhedron cannot be described as 

 any combination of axes and symmetry-proper- 

 ties of the second order, and can only be regarded as that of a 

 true mirror-axis A with a characteristic angle of 90. 



In the special case C n , where n has the value /, the symmetry of 

 the figures is the same, as when a single plane of symmetry were 

 present. Generally, therefore, the symbol S instead of C l is given 

 to this group. This symmetry plays a predominant role in the 

 description of a great number of living beings : many leaves, flowers, 

 the bodies of innumerable animals of all kinds, etc., possess this 

 symmetry. In fig. 56 the crystalform of potassium-tetrathionate: 

 /C 2 S 4 6 is reproduced, the plane of symmetry being placed here 

 in a vertical position. 



Fig. 53. 

 Dioptase. 



Fig. 54. 



