Symmetry 153 



around the stem from each other no leaf would ever be directly over any 

 below it. The advantage sometimes suggested for this arrangement, that it 

 would distribute the leaves most evenly to the light and thus be most 

 efficient in preventing shading, is open to many objections. 



The fraction 0.38197 is of interest in another connection, for it desig- 

 nates the "golden mean," or sectio aurea, the distance from the end of a 

 line at which, if the line is cut there, the smaller fraction of the line is to 

 the larger as the larger is to the whole. Thus 0.38197 : 0.61803 = 0.61803 : 

 1.0. The golden mean has long been known and has received much atten- 

 tion from artists and mathematicians, and its significance in the geometry 

 of symmetry may be considerable, but its biological importance seems 

 negligible. One should also remember that there are other series of frac- 

 tions which converge to the same limit. 



This analysis of the genetic spiral assumes that, as it twists around the 

 stem, a given leaf position on it is directly over one below, after passing 

 3, 5, 8, 13, etc., leaves on the spiral. Thus there should be vertical rows 

 of leaves, relatively few in the simpler phyllotaxies but more numerous 

 in the complex ones. These have been called orthostichies and mark the 

 end points of each successive fraction into which the genetic spiral is 

 divided. Their presence is essential if the mathematical analysis of this 

 spiral, going back to the work of Schimper and Braun and elaborated by 

 so many botanists since then, is to mean very much. 



The existence of these orthostichies, however, has been challenged by 

 more recent students of phyllotaxy, who have approached its problems 

 not by an analysis of mature structures but by a more truly morphogenetic 

 investigation of the way in which the leaves originate. The best place to 

 study leaf arrangement, they maintain, is in the bud or at the apical 

 meristem. Church (1920), one of the pioneers in this method of attack, 

 discovered that in the arrangement of primordia as seen in a cross section 

 of the bud there are no orthostichies at all, for no leaf primordium arises 

 directly over one below. Thus doubt was cast on all the early conclusions- 

 based on the assumption that the genetic spiral could be divided into re- 

 peated portions. 



But other relationships are more important than this. A study of leaf 

 primordia packed into the bud, or of other cases such as pine cones and 

 sunflower heads where there are a great many structures spirally arranged 

 but crowded together, shows the existence of another series of spirals, 

 resembling the genetic one in certain respects but reached in a different 

 fashion. If one looks at the cross section of a bud, or the face view of a 

 sunflower head in fruit, or the base of a pine cone, he will notice that the 

 units are not packed uniformly together like the pores of a honeycomb. 

 Instead, the various structures— leaves, primordia, fruits, or scales— form 

 two sets of spiral curves, starting in the center and moving to the circum- 



