CHAPTER III. 



Equivalent and N on- equivalent Operations. Definition of Group. 

 The Problem of the possible Classes of Symmetry as a Problem of 

 the Theory of Groups. The Axial Groups. Some general Theorems. 

 Cyclic, Dihedral and Endospherical Groups. -- Symmetry -Centre 

 and Enantiomorphism. 



1. In this and the next chapter it is proposed to investigate 

 what combinations of symmetry-properties are possible in the case 

 of finite figures and objects, and in what way the various types of 

 symmetrical figures can be classified. Attention was drawn in the 

 previous chapter to the important fact, that not every combination 

 of symmetry-properties can exist, because the simultaneously existent 

 symmetry-properties of a certain figure are always connected by 

 special relations which allow of a mathematical treatment of the 

 problem considered. With reference to the previous chapters, the 

 reader will remember what has been said about the equivalence of 

 symmetrical operations. In the same sense we shall speak in future of 

 equivalent operations, as being those that bring a figure from its origi- 

 nal position into the same final position. Thus e. g. if a regular hexa- 

 gonal pyramid be rotated about its vertical axis through an angle of 

 60, it will coincide with itself, because it now occupies the same part 

 of space as before ; but notwithstanding this, its new position is diffe- 

 rent from its initial one, and the same is the case if rotations around 

 the same axis are executed through angles of 1 20, 1 80, 240, and 300 

 If rotated through 360, the figure arrives, however, in the same posi- 

 tion in which it was originally. All these rotations except the last ones, 

 therefore, are non-equivalent motions, quite independent of the fact, 

 that by each of them the pyramid will be brought to self -coincidence. 



If a symmetrical figure possesses an axis of the first order with 



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