162 



sunflower (Helianthus annuus) (fig. 128), the multiple fruit of 

 Ananassa sativa (fig. 129) with its consolidated mass of berries and 

 their bracts round the axis, and finally the phyllotaxis of Euphorbia 



Wulfeni, according to Church. 

 The number can be easily 

 augmented. 



Such a periodical arrange- 

 ment evidently possesses the 

 characteristics of a space- 

 lattice wound upon a cylin- 

 dricar surf ace. There are thus 

 definite translations, by which 

 the fundamental space-lattice 

 is determined as by a special 

 kind of symmetrical opera- 

 tions. If rolled round the 

 cylindrical surface, the diver- 

 gence of consecutive leaves 

 on the genetic spiral (dotted 

 line) may be expressed by a 

 fraction, the values of wh'ich 



as found in nature l ) belong, among others, to the remarkable 

 series: |, 1* -|, |, -f^, -fa, -J|, etc.. Each fraction therein is obtained 

 from both the preceding by addition of their numerators and 

 denominators respectively. The series of these numbers was already 

 studied by Leonardo Pisano (Fibonacci; 1180 1225), by 

 Kepler, Lame, Bravais, and other mathematicians. More especial- 

 ly it may be remembered, that these fractions represent the 



There occur also divergencies in nature, the value of which belong to the 



1 

 i: ' 



T' 2x+ 1' 3x+ 

 terms of the continuous fraction: 



' - -^ -- -, etc., which may be expressed as the successive 



x+ 



1 + etc. 



Such divergencies are the rarer, the greater the value of x is. The more general 

 expression for the occurring divergencies, published by Wiesner, is : 2 _ ^ _ ^ 



