MATHEMATICS AND ENGINEERING IN NATURE 455 



easily be proved that the best distribution of the elements is obtained 

 when the lines appearing in the configuration of the flower (Fig. 5) 

 are logarithmic spirals. 5 



In a similar manner the location of the leaves along the stem of a 

 plant is determined by remarkable numerical relations. 6 The fraction 

 n/m expressing the parts of the circumference by which consecutive 

 leaves are separated are constant for each species and are the successive 

 approximations of the continued fraction 



1 



1 + 1 



1 + 1 



!+••-, 

 i. e., 1, -J, §, §, §, T %, • • •, which, on the other hand, are also the terms 

 of a special Lame's series. Arranging the leaves according to these 

 fractions, nature insures for each species the best distribution of light 

 and masses, and consequently the best growth. 



Helical motion in the growth of certain plants around supporting 

 poles may be explained mechanically. The outer parts exposed to the 

 light grow faster than the inner portions, with the resulting tendency 

 to bend the stem around the pole. This, combined with the upward 

 growth, produces the spiral motion. The fact that most climbing 

 plants, with a few exceptions (hops, Polygonum convolvulus, etc.), 

 show left-handed twisting may be circumscribed by calling it " geo- 



FlG. 5. 



6 Emch, " L 'Enseignement Mathematique, " Vol. 7, p. 722 (1910). 

 8 Strassburger, "Lehrbuch der Botanik fur Hoehschulen, " pp. 31-35. 



