Symmetry 155 



eighth, or thirteenth primordium. These are not to be seen. Furthermore, 

 if one carefully studies the angular divergence between successive pri- 

 mordia he finds (in the great majority of cases) that it is close to the 

 "ideal" Fibonacci angle of 137.5° which the series of phyllotactic fractions 

 approaches as a limit. 



The number of clockwise and of counterclockwise parastichies in a 

 given axis is not the same. In different types, however, their relative num- 

 bers are fixed and specific. These also fall into a characteristic series. Thus 

 in the bud section shown in Fig. 7-3 one can count five parastichies turn- 



Fig. 7-3. Cross section of apical bud of Pinus pinea showing absence of orthostichies. 

 The primordia, numbered in succession, are separated by the Fibonacci angle. Five 

 counterclockwise parastichies and eight clockwise ones are evident. (From R. Snow, 

 courtesy of Endeavour. ) 



ing to the left and eight to the right. In simpler forms there may be three 

 in one direction and five in the other. In more complex cases, such as 

 many pine cones, there are 8 of one and 13 of the other, or 13 of one and 

 21 of the other. Some systems have 21 and 34. Most sunflower heads show 

 34 spirals in one direction and 55 in the other. Arranging these pairs of 

 numbers in the form of fractions, as was done with the genetic spiral, one 

 obtains the series %, %, %, 8 / 13 , 13 / 2 i, 21 / 34 , 3 %5, 5 %9, and so on, though 

 the higher fractions are rare. The numbers in numerators and denomi- 

 nators form a series, as in the genetic spiral, but here the denominator of 

 one fraction forms the numerator of the next one instead of the next but 



