158 



intimately connected with all questions about ideal visual beauty 

 of proportion in art and natural forms l ). 



The "ideal" arrangement in phyllotaxis, towards a "tendency" 

 in living nature appears to exist, should therefore be considered such 

 that a spiral arrangement is attended to, whose characteristic angular 

 divergence is equal to TT (3 1/5), i. e. to 137 30'28". In this case 

 true "orthostichies" do no longer exist, because there can never 

 be a leaf standing exactly above some other, except in infinity. In 

 the opinion of the adherents of this theory, the "ideal" disposition of 

 leaves about a cylindrical stem aimed at by nature, would therefore 

 be such as to prevent each leaf from overlapping another, even if 

 the plants were so closely packed together as is often the case in dark 

 tropical forests. The question, in how far this teleological view must 

 be considered as being a mere fiction, or in real agreement with 

 the natural adapation of the plant to its need of light and free air, 

 may be passed over here 2 ). 



30. If the theory of phyllotaxis just explained be once adopted, 

 and if the forms of the parastichies on a cylindrical stem be supposed 

 to agree with that of the ordinary Archimedian spirals (p = a.s), 

 the development of the system of parastichies on the cylindrical 

 surface in a plane will give a system of parallelogram-shaped 

 meshes, at the corners of which are placed the different leaves. 

 Such a plane drawing, showing the arrangement of the bracts and 

 berries observed in a part of the multiple fruit of Ananassa sativa, 

 is reproduced in fig. 128. 



As Wulff 3 ) first pointed out, the distribution of these organs 

 is, at least in principle, exactly comparable with the space-lattice- 

 structure met with in crystalline matter, this form of structure 



1) Let a straight line AB be equal to unity, and C be a point so situated on 

 it, that AC : CB = AB : AC. Then AC 1 - = AB. BC, from which follows that 

 BC = | (3 ]/5), and AC = i (1/51). This division of AB by the point C 

 is called the "golden section", or "aurea sectio" (also: ssctio divina or divina 



proportio (Kepler)), the length of both portions is 0,381988 and 0,618034 



respectively. The relations of this ration to the properties of the regular pentagon, 

 and therefore to pentagonal symmetry in general, are well-known. 



2) J. Wiesner, Flora (1875), p. 115, 139, 142; Biol. Centralblatt. 23. 209, 

 249, (1913); H. Winckler, Pringsheim's Jahrbuch f . wiss. Botan. 36. 1. (1901). 

 Wiesner concludes: "Regular phyllotaxis as determined in the sense described 

 above, is a phenomenon doubtless intimate lyconnected with the question of the 

 most suitable adaptation to the natural conditions of light-absorption by plants". 



3 ) G. W. Wulff, Symmetry and Its Manifestations in Nature, (Russian)^ 

 Moscou (1907), p. 119. etc. 



