156 The Phenomena of Morphogenesis 



one. The fraction which this series approaches as a limit is 0.61803, the 

 larger one of the two which are separated by the golden mean. This frac- 

 tion is thus the difference between 1.0 and 0.38197, the limit approached 

 by the other series. The two spiral systems are evidently related but just 

 how they are is a nice mathematical problem. It is no wonder, as D'Arcy 

 Thompson says, that these various relationships have long appealed to 

 mystics and to those who seek to square the circle or penetrate the secrets 

 of the Great Pyramid! 



Parastichies are present in shoots around which leaves are borne in a 

 phyllotactic spiral, but because they are pulled out so far lengthwise 

 they are much less conspicuous than when many structures are packed 

 together. In elongate shoots orthostichies, though absent in buds, can 

 usually be demonstrated. The tensions resulting from elongation ap- 

 parently operate to straighten out the spirals and in many cases to bring 

 the insertion of a leaf almost directly over one that is three or five or 

 eight leaves below. Not much of a twist is needed to accomplish this and 

 to produce an orthostichy. One should recognize, however, that such are 

 secondary rather than primary phenomena of symmetry. 



The problems of phyllotaxy were already involved enough when a 

 French botanist, Lucien Plantefol (1948), added a further complexity. 

 His theory has been extensively developed by others, particularly in his 

 own country. It is based on a study of the insertion of the leaf traces on 

 the stem. Plantefol does not regard the genetic spiral as significant. He 

 traces two (sometimes more) foliar helices connecting the leaf bases in 

 parallel spirals that wind up the stem, and he usually represents these 

 helices as projected on a plane where their relationships can more easily 

 be seen (Fig. 7-4). They originate in the traces of the two cotyledons, 

 and the series remain distinct as they pass up into the bud. Here they 

 terminate in a generative center of embryonic tissue just below the tip 

 of the meristem ( Fig. 7-5 ) . In this the new primordia are differentiated. 

 The position of each is determined, he believes, by stimulation from the 

 foliar helices below, the relations being harmonized by an "organizer." 

 Lance (1952) found in a number of cases a zone of abundant mitoses 

 somewhat below the apex of the meristematic dome but few at the very 

 tip. Crockett (1957) finds some evidence of the same thing in Nicotiaha. 

 Loiseau ( 1954 ) cut off the tip of the meristem in Impatiens and observed 

 that in many instances this resulted in changing the number of helices. 

 This he believes was due to a disturbance of the generative center. 

 Popham (1958), on the other hand, in a census of mitoses at the apex 

 of Chrysanthemum, found no evidence of a generative center nor of its 

 necessary corollary, a region of few mitoses at the very tip. Newman 

 (p. 60) made the same observation in living material. The problem has 

 been discussed by Wardlaw (1957b), who concludes that, although there 



