34 



cases in which the symmetry of the figures considered is charac- 

 terised by the mere existence of axes of symmetry of the first 

 order, i. e. by mere rotations. Such figures and objects therefore must 

 always be different from their mirror-images; as we shall see later, 



--A, 



Fig. 19. 



d-Camphor-oxime. 



this kind of symme- 

 try plays an impor- 

 tant role in many 

 phenomena observed 

 in the domain of 

 chemical and physi- 

 cal sciences. 



I. The simplest 

 cases are obviously 

 those where only one 

 axis of the period 



n 



Fig. 20. 



Sodium-periodate. > 



exists. The corresponding symmetry-groups contain n non- 

 equivalent rotations, as mentioned before. We shall call them cyclic 

 groups, and indicate them 

 by the symbol O, where 

 n may have any value 

 from /to co. *) 



As instances of symme- 

 trical figures and objects 

 of this kind, in fig. ip, 20, 



and 21, the crystal-forms of optically active 

 camphor-oxime : C 10 H 16 NOH, of sodium-periodate : 

 \-\-jH 2 0, and of wulfenite : PbMoO^ are 

 reproduced; the values of n 

 are here 2, j, and ^respecti- 

 vely, and the axes are all placed vertically, 

 with the exception of that of camphor-oxime, 

 this being in horizontal position. 



Many parts of plants and animals possess 

 this cyclic symmetry, as fig. 22 to 24. convin- 



'cingly show, where the blossom-diagrams of 

 Blossom-diagram of . 



Paris quadrifolia. Paris W^olia (fig. 22; C 4 ), ') the fruits of 



Chlamydia tenacissima (fig. 23 ; C 3 ) and of Helic- 



1) For n = oo we have, properly speaking, no longer a finite group of rotations. 

 This case will therefore be considered later on more in detail. 



-) The ternary symmetry is - generally found in Monocotyledons, and Paris 



Fig. 21. 



Wulfenite. 



