53 



n 1 ci/si> he represented us resulting from the existence of an axis <>/ (lie 

 order . 1 - \i-ith a period-number -, combined u'ifli a symmetry-centre. 



n 

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



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

 A- -HUH- illustratidns of figures and objects having the symmetry 

 tin- groups C 6 , C 3 , and C 4 respectively, we give here in fig. 51, 

 and ?, the images of some polyhedra. The first represents the 

 stal-form of dioptase: CuH z SiO t , and it is at once seen that the 

 . 1,; 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 

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

 first order A 3 were present with a reflecting 

 me perpendicular to it. In the same way it will 

 obvious that in fig. 53 the special sym- 

 letry of the polyhedron cannot be described as 

 ly combination of axes and symmetry-proper- 



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

 ic mirror-axis A. } with a characteristic angle of 90. 

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

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

 snt. Generally therefore the symbol 5 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, 

 tlu- bodies of innumeral animals of all kinds, etc., manifest this 

 symmetry. In fig. 54 the crystal-form of potassium-tetrathionate: 

 K 2 S 4 6 has been reproduced; the plane of symmetry being placed 

 IK 10 in a vertical position. 



Fig. 51.. 



Dioptase. 



Fig. 52. 



