165 



in living nature, and that there is thus rather a tendency towards 

 the irrational ratio of the "aurea sectio" in living nature, which 

 to some extent goes parallel to the preference for true pentagonal 

 symmetry stated in the preceding chapters of this book. 



Wulff tries to show that this contrast is only an apparent one, 

 pointing to the fact that even in such a crystallographically admis- 

 sible space-lattice, irrational "limit "-values might be indicated. J ) 

 If a straight line, for instance, joins the angular point of such a net- 

 plane, which has the coordinates (<?,/), with the origin (= o, o), 

 it passes alternately above and beneath the angular points (5,/j), 

 (3,8), (2,5), etc., of the net-plane; and it approaches the closer 

 towards these points, the further distant the original point (m, n) was 

 chosen from 0, that original point being determined, for instance, 

 by coordinates like m = 34, n = 89, etc. The angular points men- 

 tioned have the coordinate-fractions characteristic for the consecutive 

 terms of the Fibonaccian series; and the straight line considered 

 will, in infinity, pass through the point having the tn-ational coordi- 

 nates: Af[(3-l/5),2). However, it is clear that there is in the whole 

 infinitely extended net-plane, no such point really present; and 

 it cannot be maintained that a parallelism between the space-lattice- 

 character of crystals with their rational indices, and between living 

 organisms is really established by this mode of reasoning, suggestive 

 as it may be for the remaining. For the generatrix of the cylindrical 

 stem would in this case be a line of the supposed space-lattice, which 

 does not pass through any real angular point of the net-plane consi- 

 dered; this straight line would, therefore, have no significance at all 

 in a crystalline medium. Instead of supporting such a parallelism 

 between the two groups of phenomena, the fundamental contrast 

 between living and inanimate nature with respect to form-symmetry, 

 is again more strikingly brought to the fore by these considerations 

 in so far, as what is impossible in the one domain of phenomena, 

 should even be the most perfect state of things in the other. 



31. However, it must be borne in mind that we have no right 

 to consider these remarkable views on phyllotaxis, as pictured in 

 the above, to be of real ontogenetic significance, as long as we have 

 no indications about the mechanical or physiological causes of such 

 mathematically determinable arrangements of similar organs. 



In this respect it is now of importance to remark, that the cor- 



l ) G. W. Wulff, loco cit, 



