132 



repeats, the symmetry of the resulting pattern will not be influenced 

 by the specific geometrical nature of these repeats, if the latter only 

 be all identical and placed in an analogous way with respect to 

 the axes of the system. 



But if these patterns should also have symmetry-properties of 

 the second order, e. g. a centrical symmetry or symmetry-planes, 

 then again special symmetry-properties of this kind must be attribu- 

 ted to the motifs themselves, just as appeared to be the case in 

 Bravais' explanation of the lower symmetrical crystal- forms. 

 Sohncke's theory shows, therefore, in this respect an analogous 

 deficiency to that of Bravais, if used for the explanation of such 

 higher symmetrical, crystal-structures, although its deficiency has 

 another significance, and is not so strongly marked, as that of 

 Bravais' view. Therefore Sohnckes' theory must certainly be 

 considered to be a real progress in comparison with that of the 

 latter, although the problem mentioned above has evidently not 

 yet got its most general solution by it. 1 ) 



12. Before finishing these considerations of Sohncke's regular 

 systems, we may remark here, that the theory can be extended 



also to cases in which the constitu- 

 ^ ve re P eats f t ^ ie tridimensional 

 pattern are no longer of the same 

 kind, but of different character. If a 

 definite number of such Sohnckian 

 systems, which all possess the same 



^ ^ and parallel translations, but which 

 f \*J \J/ ^5 are neither congruent nor need be 

 * built up of the same particles, be 



Fig. 118. suitably placed the one into the 



other, such an interpenetration can 



lead to a complex, materially heterogeneous system, the foundation 

 of which is a space-lattice which is characterised by the trans- 

 lations just mentioned. 



As an instance of this, a section of such a periodical pattern has 

 been represented in fig. 118 . It is deduced from the pattern of fig. 

 115 in such a way that a motif of an other kind is placed every time 

 at the centre of each hexagonal group of fig. 775. The fundamental 



:; :6; :; 







l ) Cf. also: P. Niggli, "Geometrische Krystallographie des Diskontinuums" , 

 Leipzig, (1918, 1919). 



