127 



morphously related kinds of repeats, and only such patterns as are 

 themselves different from their mirror-images, i. e. which possess 

 only symmetry-properties of the first order, are in general formed 

 by the regular arrangement of one and the same kind of pattern-unit. 



The regular structures, as deduced by Sohncke, are completely 

 determined by rotations and translations; the latter and their com- 

 binations with certain motions about axes of the first order, which 

 represent therefore helicoidal motions, -- are indeed operations of 

 essential significance for unlimited systems, as we have seen in 

 Chapter //. 



Owing to the fact that in these unlimited systems there are sets 

 of parallel axes of rotation or helicoidal motion, it is of interest to 

 point here again to the fact that the simultaneous existence of such 

 parallel axes always involves * A 



the existence of others, which v* / _ T X 

 can be found by the construc- 

 tion of Euler (see Chapter //, 

 p. 29). Some examples may 

 make this clear. 



Let (fig. in) A l and A 2 be 

 two parallel quaternary axes. 

 If we apply Euler's construc- 

 tion to find the resulting axis, 

 we must realise, that the centre of the sphere used in fig. 20, is now 

 at infinite distance, the surface of the sphere, therefore, being changed 

 into a plane perpendicular to A l and A 2 , and thus coinciding with the 

 plane of our drawing here. When the rotations are both clockwise, 



we must construct the angles - - (= 45) as indicated in the 



figure, and because /_A^A^A^ = 90 therefore, it appears that A 3 

 is a binary axis (^ = 180), parallel to A l and A 2 . Indeed, the exis- 

 tence of such parallel binary axes, as a necessary consequence of 

 the presence of A l and A z , is confirmed, for instance, by the patterns 

 of fig. 108, 109, 114, etc.; the arrangement of the quaternary axes 

 of the pattern appears the same as that of the alternating binary 

 axes. In the same way it is seen from fig. 115, that the senary axes 

 alternate with sets of ternary and of binary axes there, which follow 

 from the simultaneous presence of the parallel senary axes in exactly 

 the same way. 



If, however, the rotations round A 1 and A 2 had opposite directions, 



