116 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS. 



be considered, the process being rendered obvious by the fact that 

 the ray-florets in the case of the Sunflower occupy the transitional 

 areas between two cycles. 



Comparison of the figure in which 21 ray-florets are indicated as 

 black patches shows that these follow a remarkable sequence which, 

 counting from No. 1 in the direction of the genetic spiral, may be 

 represented by the figures — 



2-1-2-1-2 I 2-1-2-1-2 I 2-1-2 



The fact is thus brought out that the essence of the Fibonacci 

 series consists in the manner in which it may be regarded as com- 

 posed of the expression 2-|-l-f-2-Kl-f2 treated as a recurring 

 quantity. Thus 3 of these members add up to 5, 5 to 8, 8 to 13, 

 any 13 to 21, and any 21 to 34, etc. Any ratio of the series may 

 undergo subdivision in this sense to produce the next higher mem- 

 ber. From this it follows that the law of arranging members of a 

 higher cycle on a preceding lower one consists in the method of 

 dividing them in sequence in the order indicated. And conversely, 

 whenever increase of members takes place in such a manner, it is 

 at once clear that a transitional sequence of the Fibonacci series is 

 implied {cf. Cactaceae). 



To subdivide a phyllotaxis system so as to retain the Fibonacci 

 ratio, it is therefore only necessary to start from No. 1, in the 

 direction of the genetic spiral, and put in new paths in the 

 sequence (2 • 1 • 2 • 1 • 2), or graphically — 



iVIVIV 



VIVIV 



etc. 



When two such transitions are involved, the sequence becomes 

 (3 • 2 • 3 • 2 • 3), or— 



„ VI V IV V IV , 



II. . . V I V i V ' 



By noting this property, the (8 -|- 13) curves are selected from the 

 (21-1-34) set, the sequence being carried out along the direction of 

 the spiral concerned ; thus in the practical construction of diagrams, 

 it is necessary to start from 1 and proceed' from the concave side of 



