72 



Fig. 82. 



Pyrite. 



Fig. 83. 



Boracite. 



a figure can possess if no axes with n = oo are taken into account. *) 

 Finally we may draw attention to the fact that the group K H 

 contains all operations which are characteristic as well of the group 

 T H as of T D . These last are therefore called sub-groups of K H : 

 In the same way the groups T and K 

 themselves are sub-groups of K H . Also in 

 the case of the 

 other symmetry- 

 groups now dedu- 

 ced, we can indi- 

 cate such sub- 

 groups as contain 

 a part of the ope- 

 rations of other, 

 higher symmetri- 

 cal combinations 



of symmetry- 

 elements. This fact is of importance, as we shall see afterwards, 

 for the sake of combining several groups to larger ones, a process 

 which is the basis of the division in crystal-systems and crystal- 

 classes, as since early days it has been used in cristallography, 



and which simultaneously explains the 

 deeper meaning of the old division of crys- 

 tallographical polyhedra into holohedral, 

 hemihedral, and tetartohedral forms, as was 

 especially brought to the fore in Nau- 

 mann's doctrine 



12. As illustrative examples of this 

 symmetry, in fig. 82, 83, and 84, the crystal- 

 forms of pyrite: FeS 2 (fig. 82; T H ), of 

 boracite: Mg^B u Cl 2 M , (fig. 83; T D ), and 

 of fluorspar: CaF 2 (fig. 84; K H ) are repro- 

 duced as some instances of the groups T H , T D , and K H respec- 

 tively. 



Of living beings, the pollen-cells of some plants may perhaps be 

 mentioned here: thus of group T H perhaps those of Buchholzia 

 maritimci; of T D those of Corydalis sempervirens, and of group K H 



*) Of course, if axes of isotropy are concerned too, the spherical symmetry 

 is the highest possible' one. Indeed, just in the same way as the sphere is an 

 "endospherical" polyhedron with an infinite number of faces. 



Fig. 84. 

 Fluorspar. 



