157 



on the genetic spiral (dotted line) may be expressed by a 



iet ion, the values of which as found in nature 1 ), belong, among 



UTS, to the remarkable series: -J, -J-, , f, -fe, -fa, ^ - t, I-..K h 



i. tit MI thcn-in is obtained 



)m both the preceding by 



1< lit ion of their numerators 



id denominators respectively. 



le series of these numbers 

 already studied by Leo- 

 lardo Pisano (Fibonacci; 

 180 1225), by Kepler, 

 .ame", Bravais, and other 



ithematicians. More especi- 

 ly it may be remembered that 

 lese fractions represent the 

 iccessive values of the stages 



the continuous fraction: 



1 



2 + 1 



1 +1 



1 +1 



Fig. 127. 

 Pine apple. 



1 + etc., 



dues which oscillate alterna- 



ily towards the positive or negative with respect to a definite li- 

 iting number, to which the successive terms continuously approach 

 lore closely. This true limit-value is no other than the irrational 

 imber ^ (3 1/5), which represents the smaller portion of the 

 itio known as the "aurea sectio", a ratio which since the days 

 Leonardo da Vinci (1452 1519) has been considered to be 



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



series , 

 x 



1 



1' 2x+ 1' 3x+ 



terms of the continuous fraction: 

 1 



i -, etc. which may be expressed as the successive 



1 +1 



1 + etc. 

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



2* 1 V/5 



expression for the occurring divergencies, published by Wiesner, is: 



