Consider this challenge: Imagine trying to engineer a plant so that new growths are compactly arranged around the growing point with no wasted space. Suppose you chose to make each new primordium grow out at an angle of two fifths of a revolution from the previous growth. You would have the problem of every fifth primordium growing from the same spot and in the same direction. They would form rows with wasted space between the rows. (See figure 3.) The truth is, any simple fraction of a revolution results in rows rather than optimal packing. Only what has been termed the “golden angle” of approximately 137.5 degrees results in an ideally compact arrangement of growths. (See figure 5.) What makes this angle so special?

The golden angle is ideal because it cannot be expressed as a simple fraction of a revolution. The fraction 5/8 is close to it, 8/13 is closer, and 13/21 is closer still, but no fraction exactly expresses the golden proportion of a revolution. Thus, when a new growth on the meristem develops at this fixed angle with respect to the preceding growth, no two growths will ever develop in exactly the same direction. (See figure 4.) Consequently, instead of forming radial arms, the primordia form spirals.

Remarkably, a computer simulation of primordia growing from a central point produces recognizable spirals only if the angle between new growths is correct to a high degree of accuracy. Straying from the golden angle by even one tenth of a degree causes the effect to be lost.​—See figure 5.

How Many Petals on a Flower?

Interestingly, the number of spirals that result from growth based on the golden angle is usually a number from a series called the Fibonacci sequence. This series was first described by the 13th-century Italian mathematician known as Leonardo Fibonacci. In this progression, each number after 1 is equal to the sum of the previous two numbers​—1, 1, 2, 3, 5, 8, 13, 21, 34, 55, and so on.

“Everything He Has Made Pretty”

Artists have long recognized the golden proportion as the most pleasing to our eyes. What makes plants form new growths precisely at this intriguing angle? Many people conclude that this is but another example of intelligent design in living things.