110 



made to coincide with some others by the non-equivalent operations 

 of the symmetry-group to which the figure as a whole belongs. 



But occasionally we have drawn attention to the fact that there 

 are also figures in which an endless number of points may correspond 

 to any given point ; it may happen that no point of the system re- 

 mains at its place in space, should the system be subjected to the 

 set of non-equivalent operations characteristic of its symmetry. 

 Such figures are called endless, unlimited, or infinitely extended figures. 



It will be remembered that in Chapter II, several symmetrical 

 operations were considered which have no real significance for limited 

 figures, as, for instance, translations, helicoidal motions, rotations 

 about axes or reflections in planes not passing through the same point 

 in space, etc. Such operations may however be of essential interest 

 just for such wwlimited systems. 



A detailed account of the structure-theories and an exhaustive 

 treatment of the remarkable properties of all possible unlimited 

 symmetrical arrangements would be out of place here. We wish only 

 to give an impression of the most salient features of such systems, 

 and more particularly to show the importance of the views dealt 

 with, for the problem of the internal structure of crystalline matter. 

 As these views have in recent times met with most happy endorse- 

 ment from direct experiments, it seemed desirable to dwell somewhat 

 longer upon the results obtained in this way and upon the methods 

 applied in these investigations. Finally, some remarks on arran- 

 gements of this kind as met with in living nature will be made with 

 a view to drawing the attention of the reader to these applications of 

 the doctrine of the regular unlimited systems, also in questions of the 

 arrangement in space of the organs in living individuals. Even if 

 only preliminary, and giving no true explanation of the mechanical 

 and physiological causes governing the said phenomena, the views 

 about them are suggestive enough to be worthy of more detailed 

 examination in the future from the standpoint of the general 

 doctrine considered in this book. 



2. If a plane figure be repeated again and again in the plane of 

 drawing, in such a way that proceeding in some direction, we meet 

 after equal distances identical and identically oriented figures, it may 

 be said that the repetition of the original figures occurs periodically ; 

 the length of the distance between two consecutive figures in the same 

 position is called the period of the arrangement in the direction con- 

 sidered. The complete, infinitely extended assemblage thus obtained 



