74 



now deduced, we can indicate such sub-groups, as containing only a 

 part of the operations of other higher symmetrical combinations of 

 symmetry-elements. This fact is of importance, as we shall see after- 

 wards, for the sake of combining several groups to larger ones, - 

 a process which is the basis of the division in cry stalsy stems and 

 crystak/asstfs, as since early days it has been used in crystallography, 

 and which simultaneously explains the deeper meaning of the old 

 division of crystallographical polyhedra into holohedral, hemihedral, 



and tetartohedral forms, as was especially 

 brought to the fore in Naumann's doctrine. 

 13. As illustrative examples of this 

 symmetry, in fig. 84, 85, and 86, the crystal- 

 forms of pyrite: FeS 2 (fig. 84', T H ), of 

 boracite: Mg 7 B 16 Cl 2 30 , (fig. 8 5 \ T*>), and 

 of fluorspar: CaF 2 (fig. 86\ K H ) are repro- 

 duced as some instances of the groups T H , 



T D , and K H respectively. 

 Fig. 86. 



Fluorspar ^ living beings, the pollen-cells of some 



plants may perhaps be mentioned here: 



thus of group T H perhaps those of Buchholzia maritima; of T D 

 those of Corydalis sempervirens, and of group K H the pollen-cells 

 of some Polygoneae, according to Haeckel's data. 



However, it is difficult to say whether such individuals really 

 belong to this class or only have the symmetry of the groups 

 T and K themselves. If so, the drawings of fig. 51 may be included 

 here, or the instances just mentioned may be among those given 

 in the preceding chapter. 



14. No other symmetry-groups than those deduced in the 

 preceding are possible for finite stereometrical figures, as long as 

 axes of isotropy are -not concerned. The whole investigation has 

 therefore led to the result that the different types of symmetrical 

 figures are- only few in number, although of course their total number 

 is infinitely great, because n can have all possible values. 



If we review these principal types here once more, we shall find 

 the following result: 



A. Symmetrical figures which differ from their mirror-images. 



1. Cyclic groups C n 



2. Dihedron-groups: D n 



j. Endospherical groups: T, K and P. 

 All figures belonging to A may exist in two enantiomorphous forms. 



