rotated through an angle of 72 round the axis A already mentioned; 

 and this can be done five times in the same direction. After the fifth 

 motion the figure is again in exactly the same position as it was at 

 the beginning. It seems to be adequate in this case to define the 

 typical symmetry of the figure by these characteristic rotations. 

 Although in the case considered this will really appear to be justified, 

 we have, however, still to modify our definition of symmetrical figures 

 with respect to another particular, before it can seem complete. 

 In fig. 5 a complex of dots, just like those in fig. 2, is drawn in the 

 same mutual positions 

 and of the same 

 magnitude ; the figure 

 evidently possesses 

 the same symmetry 

 as the original one; 

 but in spite of the 

 fact that all details 

 and properties are the 

 same as in fig. 2, it 

 will appear to be im- 

 possible now to move 

 the figure in its pla- 

 ne l ) in such a way, as 

 to make it coincide 

 with fig. 2. We can, 

 however, obtain fig. 2 

 from fig. 5 by reflec- 

 ting the last one in a 



mirror S, placed perpendicular to the plane of the drawing; the 

 mirror-image of fig. 5 now obtained, is really congruent with fig. 2 

 itself, and it can now be brought into coincidence with it by mere 

 shifting and rotating in its own plane. Because of this relation, 

 we say that the plane figures 2 and 5 are each other's mirror- 



Fig. 5. 



*) The condition that this two-dimensional figure remains in its own 

 plane during its motions, is essential in this mode of argument. In a tridi- 

 mensional space the figure would be brought to coincidence with itself by a 

 mere rotation through 180* round an axis situated in its own plane; conf. also: 

 A. Griinwald, Die Stulpungen unseres Raumes, Prag-Bubentsch, (1914), pag.6. 

 In this paper are also concerned the corresponding relations in polydimensional 

 spaces. 



