The Golden Angle/Chapter 7

The golden angle was obtained by a group of teachers examining the sunflower leaves, then finding fractions of 360° using ratios of alternate Fibonacci numbers :

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ...

 1/2  x 360° = 180° 1/3  x 360° = 120° 2/5 x 360° = 144°     3/8 x 360° = 135°   5/13 x 360° = 138.46...° 8/21 x 360° = 137.14...° 13/34 x 360° = 137.64...°  21/55 x 360° = 137.46...° 34/89 x 360° = 137.52...° 55/144 x 360° = 137.50°

This gives us an infinite alternating sequence whose limit is the golden angle, about 137.5° or 137°30'28". This is the angle that allows each leaf to be closest to the leaf below it in the previous whorl and farthest from the youngest previous leaf above it; in other words, it allows the leaf to get maximum sunlight. This number can be written as

(.618034)^2* 360  =   * 360, and also = * 360.

An example of a plant whose leaves whorl at an angle of 144°, which is in the sequence above:

See Cristobal Vila's beautiful short movie "Nature by Numbers" at

See Lori and Don's Nautilus shell applet which shows the growth of the spiral shell from 1 to 3.2 in 360o and the constant angle of 79.5o between the tangent to the curve and the radius.