14 



clearness and reference, are numbered 1 to 8. Let us moreover take 

 an arbitrary point P in space, outside the cube A, and draw a 

 straight line LL' through it parallel to one of the four upright edges 

 of the cube. If now A be revolved round the axis LL' through angles 



of 90, 2 x90,3 x 

 90, etc., the cube A 

 comes successively 

 into the positions 

 B, C, D, which posi- 

 tions differ from A, 

 as the numbers at 

 the corners clearly 

 show. Because the 

 cube now has a spe- 

 cial symmetry of its 

 own however, the 

 figures B, C, and D, 

 can be made to coin- 

 cide eventually with 

 A , by merely shifting them parallel to themselves along the plane 

 of revolution. Thus, by definition, the rotations through angles of 90, 

 2 x 90, 3 X 90 round an axis parallel to one of the edges of the 

 cube, are characteristic rotations for the symmetry of this figure. 

 As the positions B, C, and D differ from A, these three characteristic 

 rotations are non-equivalent. A rotation through an angle of 450, 

 however, would be equivalent to that through 90, etc. 



In the same way we should find, that if through some point in 

 space Q a straight line were drawn parallel to one of the four longest 

 diagonals of the cube, rotations through angles of 120 and 2 x 120 

 round this line as an axis, would appear also to be "characteristic 

 motions". It is easily seen that for the plane drawing oifig.f, rotations 

 through 72, 2 x 72, 3 x 72, and 4 x 72 round an arbitrary axis 

 passing through a point R in space and perpendicular to the plane of 

 the drawing, are also characteristic, and non-equivalent motions. 

 In the same way it must be evident that if the rotation of the 

 cube A in fig. 8 around LL 1 had been through an angle differing from 

 those mentioned, coincidence could not have been realised by shifting 

 alone; and the same would have been the case, if the rotations around 

 LL' through 90, etc., were applied to a different figure instead of 

 to a cube, e. g. to an unsymmetrical figure. 



