14 



Characteristic motions for a symmetrical figure therefore are in 

 general those by. which the figure is brought into positions parallel to 

 the initial one, but yet differing from it. 



It is of importance to point out here, that the translations mentio- 

 ned are reduced to zero, when the point P in space is so chosen, that 



it coincides with the ,,geo- 

 '120 iao 



120X 



I8(f 



120' 



90 



160 



120 



Fig. 7. 



120 metrical centre" of the 

 cube (fig. 7). After each 

 rotation it will now occu- 

 180* py the same place in spa- 

 ce, although of course 

 always with interchanged 

 ,. corners, etc., - - just in 

 the same way as hap- 

 pened in the rotations 

 first considered. 



The symmetry of a 

 stereometrical figure may 

 now be exactly defined as the total complex of its non-equivalent 

 characteristic motions, as long as only symmetry-properties of the 

 kind here considered are dealt with. 



2. The second case that we must now consider in detail is, 

 when a figure F in a position Sj is reflected in a mirror. It is then 

 transformed into its mirror-image F' and brought into a new position 

 S 2 ' ', F' is of course now no longer congruent with F. Accordingly, 

 the manipulation required to make them coincide is no longer a 

 simple motion, some further operation being required besides it. 

 If a symmetrical figure is of such a kind that it is equal to its 

 mirror-image in several ways, then it will be always possible to 

 find for that figure a series of characteristic reflections, in the same 

 sense as we have spoken of characteristic motions. In this case too 

 the point P in space, through which the mirror-planes are drawn, 

 may be chosen in such a way that the translations by which the 

 reflected figures are finally shifted to the place of the original one, 

 are reduced to zero; the figure F then remaining in the same place, 

 but in different positions after each reflection. In the case of the 

 cube, P had to be made to coincide with the centre already mentio- 

 ned, the nine (3 + 6) possible characteristic reflecting planes all 

 passing through 0. 



As we shall soon see, a "figure being in -several ways equal to its 



