12 



The different ways in which the congruency mentioned appears, 

 determines the symmetry-properties characteristic for the stereome- 

 trical figure, and with them, the whole symmetry -character of it is 

 given at the same time. 



The complete set of symmetry-properties of every figure must 

 thus be found out, before we can say what its particular symmetry- 

 character really is. As we shall see later, however, not every arbitrary 

 combination of such symmetry-properties can occur in any special 



C C 



Fig. 7. 



case; if present together, they are evidently in some way connected 

 and dependent upon each other. In the next chapter, therefore, we 

 shall see in what way symmetry-properties can be generally defined, 

 and what is the special mutual dependency of them, if more of them 

 are simultaneously present. 



quite sufficient for the complete mathematical deduction of all possible symme- 

 trical systems, as Von Fedorow and Schoenflies have demonstrated. The 

 old definition of Mobius is free from this dualism. It says: ,,Zwei Figuren heissen 

 einander gleich und ahnlich, wenn jedem Punkte der einen Figur em Punkt der 

 anderen dergestalt entspricht, dass der gegenseitige Abstand je zweier Punkte 

 der einen Figur, dem gegenseitigen Abstande der zwei entsprechenden Punkte 

 der anderen Figur, gleich ist. Es gibt aber Figuren, welche sich selbst auf mehr 

 als eine Art gleich und ahnlich sind ; . . . . solche Figuren sollen symmetrisch genannt 

 werden". Reflection in a mirror will really preserve the original relations and 

 distances of the different points to each other also in the mirror-image. This can 

 easily be demonstrated; cf. also: A. Grunwald, Die Stulpungen unseres Raumes, 

 Prag-Bubentsch, (1914), pag. 5, 6, 7. 



