Symmetry 161 



in recent years the Snows have brought forward evidence in its favor. 

 Among other experiments ( 1952 ) they isolated by two radial cuts the 

 larger part, but not the whole, of the area presumptively to be occupied 

 by the next-but-one leaf primordium. In such a case none develops be- 

 tween the cuts, although this region grows and continues otherwise to 

 be normal. They explain this result as due to the fact that the area now 

 available was too small for a primordium to be formed. 



These two hypotheses, though stressing different factors, are not dia- 

 metrically opposed to each other. What is to be explained is the even 

 distribution of primordia, equidistant from each other (in origin) and 

 regularly arranged. This is the same problem posed by the distribution 

 of multiple structures. Something regulates the differentiation of each of 

 these structures in such a way that each occupies an area of its own and 

 that these individual areas are of about the same size. In the case of 

 leaf primordia the situation is complicated by the fact that these arise 

 on a curved surface and in a progressive series in time. Although me- 

 chanical and chemical factors are doubtless involved in the distribution 

 of primordia, as in all morphogenetic processes, it is perhaps too 

 simple an explanation to regard the determination of each as due to 

 crowding by its neighbors, to the presence of the "first available space," 

 or to inhibition by other primordia. It seems more logical to regard the 

 problem of the distribution of primordia at the growing point as another 

 instance of a self-regulating biological pattern which may have its roots 

 in genetic factors, the fine structure of protoplasm, or whatever else may 

 be responsible for organic form. 



On either hypothesis mentioned above, if primordia are to arise in a 

 spiral around the axis each should be as far as possible from its immediate 

 neighbors, those coming just before and just after it in origin. In op- 

 posite leaves each is placed as far away as possible, 180°. In spiral 

 phyllotaxy this cannot be done. If primordium B, let us say, originates 

 at an angle from A of 137.5° (the golden-mean fraction of the circum- 

 ference), and if the next one, C, is placed at the same distance farther 

 on (thus incidentally dividing the remainder of the circumference by 

 the same ideal proportion), B is equidistant from A and C, and this is 

 the maximum possible distance at which successive members can be 

 placed from each other. If the distance A-B and B-C is less or greater 

 than this ideal angle, C will not arise in the middle of the largest space 

 available, as Hofmeister's postulate requires. What this means is that 

 only if successive primordia are separated by this ideal angle will they fill 

 the available space evenly and with the greatest economy. This is the 

 property of golden-mean spacing that makes it significant in problems 

 of this sort. 



Richards (1948, 1950) has worked out some of the implications of 



