44 



it; and fifteen binary axes having the directions of the lines joining the 

 middles of every two. opposite edges; all these axes are homopolar. (fig. 4.3). 

 9. Finally it may be remarked that there exists an important 

 theorem dealing with the number of non-equivalent, characteristic 

 operations, making all symmetrical figures of these groups coin- 

 cide with themselves. For the group T this number is evidently: 

 1 + 3 _j_ 4 x 2 = 12; for the group K: 1 +3 x 3 + 4x2 +6 = 24; 

 for the group P: 1 + 6 x 4 + 10 X 2 .+ 15 = 60; the rotation 

 through 360 is of course only counted once here. 



Now the number of these non-equivalent operations is in every 

 case = 2X, where x indicates the number of the edges of the tetrahedron, 

 cube, or pentagonal dodecahedron respectively. 



Indeed it appears to be a general property of each regular polyhedron 

 with x edges, that it can be brought to self-coincidence in 2X different ways. 

 This theorem is easily and quite generally demonstrable. It is 

 connected with the simple fact that every edge AB, by interchange, 

 can be placed so that its end A coincides with A or with B of any 

 other edge present. 



10. Figures and objects of this kind are represented in fig. 44, 



and 45. They 

 give the crys- 

 tal-forms of 

 barium-nitrate: 

 Ba(N0 3 ) 2> and 

 of cuprite : 

 Cu z O, from 

 Cornwall, as 

 illustrations of 

 the symmetry 

 of the groups 

 The symmetry 



P 



Fig 44. 



Barium-nitrate. 



Fig. 45- 

 Cuprite. (Cornwall). 



T and K respectively. 



of the group P is not possible in the 



domain of crystalline matter, for a reason to be explained later 



on. Of course it needs no detailed discussion to see, that every figure 



of this kind may exist in a second form which is the mirror-image 



of the other one. 



In the domain of living objects, a number of highly remarkable 

 instances of all three groups T, K, and P, are known. Indeed there 

 is nothing to prevent the occurrence of the symmetry of group P, 

 as there is with solid matter. 



