924 ON LEAF-ARRANGEMENT [ch. 



fraction is the numerator of the next; and further, that each suc- 

 cessive numerator, or denominator, is the sum of the preceding two. 

 Our immediate problem, then, is to determine, if possible, how these 

 numerical coincidences come about, and why these particular numbers 

 should be so commonly met with as to be considered "normal" and 

 characteristic features of the general phenomenon of phyllotaxis. The 

 following account is based on a short paper by Professor P. Gr. Tait*. 

 Of the two following diagrams, Fig. 450 represents the general 

 case, and Fig. 451 a particular one, for the sake of possibly greater 

 simplicity. Both diagrams represent a portion of a branch, or fir- 

 cone, regarded as cylindrical, and unwrapped to form a plane surface. 

 A, a, at the two ends of the base-line, represent the same initial 

 leaf or scale; is a leaf which can be reached from ^ by m steps 



O 



Fig. 450. 



in a right-hand spiral (developed into the straight line AO), and by 

 n steps from a in a left-handed spiral aO. Now it is obvious in our 

 fir-cone, that we can include all the scales upon the cone by taking 

 so many spirals in the one direction, and again include them all by so 



speculating on the same facts, put forward the curious suggestion that the cellular 

 tissue of the dicotyledons, or exogens, would be found to consist of dodecahedra, 

 and that of the monocotyledons or endogens of icosahedra {On the mathematical 

 connection between the parts of vegetables: abstract of a Memoir read before the 

 Royal Society in the year 1811 (privately printed, n.d.). Cf. De Candolle, Organo- 

 genie vegetale, i, p. 534. See also C. E. Wasteels, Over de Fibonaccigetalen, 

 Me Natuur. Congres, Antwerpen, 1899, pp. 25-37; R. C. Archibald, in Jay 

 Hambidge's Dynawfc Fiymmetry, 1920, pp. 146-157; and, on the many mathematical 

 properties of the series, L. E. Dickson, Theory of Numbers, i, pp. 393-411. 1919. 



Of these famous and fascinating numbers a mathematical, friend writes to me: 

 "All the romance of continued fractions, linear recurrence relations, surd approxi- 

 mations to integers and the rest, lies in them, and they are a source of endless 

 curiosity. How interesting it is to see them striving to attain the unattainable, 

 the golden ratio, for instance; and this is only one of hundreds of such relations." 



* Proc. R.S.E. vn, p. 391, 1872. 



