10 



like A BCD. The cube is thus a figure, which in nine different 

 ways is equal to.its mirror-image. On the other hand, if (fig. 5} an 

 irregular tetrahedron ABCD is taken, it is easily seen to be different 

 from its mirror-image A'B'C'D'. Such a tetrahedron therefore is an 

 instance of a figure, which is in no way equal to its mirror-image. 



6. We have dwelt somewhat long on this matter, because in 

 the beginning it appears often difficult to obtain a complete and 

 clear insight into these relations which on the other hand must be 

 considered as of fundamental significance for the following. - 



Now there are innumerable stereometrical figures which are 

 similar to themselves as mirror-images: the cube, the octahedron, 

 the cone, etc., are all instances of this kind. They all possess a 



C C 



Fig. 5- 



property which fig. 5, and fig. 2 and J, if considered in their own 

 plane, evidently lack. However, figures such as fig. 2 or 3, are 

 surely "symmetrical" figures, and a fortiori, or at least for the 

 same reason, the figures mentioned above must be called "sym- 

 metrical" ones too. 



Therefore our previous definition of a "symmetrical figure" needs 

 an extension in such a way, that we say: symmetrical figures are 

 such as are similar to themselves or to their mirror-images in more than 

 one way. l ) In more than one way: for every figure is at least equal 

 to itself by mere identity. 



J ) Indeed, there is a dualistic character in this definition, although it is 

 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 ein 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 



