152 The Phenomena of Morphogenesis 



spiral— represents the order in which the leaf primordia are formed in the 

 bud. Their positions in the spiral are not indefinite but commonly fall into 

 a few precise categories, the relations of which have long excited the 

 interest of morphologists. Simplest of all is the distichous, or truly alter- 

 nate, arrangement, with successive leaves 180° apart around the stem. To 

 pass from a leaf to one directly above it involves one circuit of the axis 

 and two leaves, a condition which may be expressed by the fraction Vo. In 

 other types this spiral passes once around the axis but every third leaf is 

 over one below it, a condition that may be represented by the fraction %. 

 Commoner than either of these is a spiral where in passing from a leaf to 

 one above it two circuits of the axis are made and the fifth leaf is reached, 



Fig. 7-1. Diagram showing % phyllotaxy. 



the % type. Frequently observed in stems is a % phyllotaxy (Fig. 7-1) 

 and less commonly that of % 3 . In cones and other compact axes more 

 complex phyllotaxies of % t , x % 4 , and 2 % 5 may be found. The series is 

 thus V 2 , Vs, %, %, % 8 , % lt 13 / 34 , 2 %5, 3 %9, and so on. Each obviously 

 represents the fraction of the circumference of the axis, or the angle, 

 traversed by the spiral in passing from one leaf to the next. The number 

 in both numerator and denominator of each fraction is the sum of those 

 in the two preceding fractions. This particular series is known as the 

 Fibonacci series. The higher fractions become more and more uniform 

 and approach as a limit the decimal fraction 0.38197, or the angle 

 137°30'28", the so-called "ideal" angle. It has been shown by Wright 

 ( 1873 ) that if successive leaves were formed at just this angular distance 



