Symmetry 163 



is likely to be flatter (as in a cone or flat head), the primordia will be 

 more numerous, and the parastichy fractions will have higher numbers. 



Richards calls attention to the fact that the parastichies at a growing 

 point are not limited to the two conspicuous "contact" ones emphasized 

 by Church but that there may be a series of others though these are 

 not obvious since they do not intersect each other at right angles. An 

 advantage of this concept is that it makes clear how the parastichies shift 

 from one pair of numbers to another, a problem that has troubled stu- 

 dents of phyllotaxy. Richards (1948; Fig. 7-7) has constructed a diagram 

 of a rather large meristem, something like a sunflower head, showing a 

 series of primordial positions numbered along the genetic spiral in 

 which each diverges from the last by the Fibonacci angle of 137.5°. In 

 such a system one can readily trace parastichies. Near the center there 

 are five counterclockwise ones crossing eight clockwise, the % arrange- 

 ment, which intersect at approximately right angles. To trace this series 

 very far out becomes difficult since the angles of intersection diverge in- 

 creasingly from 90°. As one moves out, therefore, the system seems to 

 change and the five counterclockwise spirals shift to thirteen, giving the 

 % 3 arrangements of spirals that now have more nearly right-angled 

 intersections. Still farther out the eight clockwise spirals are less easy 

 to trace, and 21 others become more conspicuous, now making the 1 %i 

 arrangement and restoring the steeper intersections. Thus in the more 

 complex systems with large, flat meristems and little difference in radial 

 distance between successive primordia, the parastichies, at least those 

 that are conspicuous and easy to trace, may be seen to shift to pro- 

 gressively higher numbers. This does not happen in ordinary shoots 

 where the meristem is steeper and the primordia are fewer and larger 

 and increase rapidly in size at each plastochron but it may sometimes be 

 seen even in such cases (Fig. 7-8). These changes involve no biological 

 mystery, as Church was inclined to believe they do, but are simply the 

 result of the unique properties of the Fibonacci angle. 



Barthelmess ( 1954 ) has pointed out that the scheme proposed by 

 Richards is essentially a two-dimensional one, whereas the meristematic 

 region has three dimensions, a fact that must be taken into account. 

 There are various other complications presented by an analysis of 

 phyllotactic patterns. Bilhuber (1933) and others, for example, find 

 that the situation in many of the cacti is often different from that in 

 most families. These plants are essentially leafless and have angled 

 stems so that in the apical regions one actually finds what look like 

 orthostichies, which are related to the development of the angled pat- 

 tern. Bijugate spirals (Hirmer, 1931; Snow, 1950) occur in some groups, 

 where a % pair of parastichies, for example, becomes split into a % . 

 Here primordia occur in opposite pairs but the plane of each pair is not 



