The Fibonacci Sequence In Nature
The Fibonacci sequence appears in unexpected places such as in the growth of plants, especially in the number of petals on flowers, in the arrangement of leaves on a plant stem, and in the number of rows of seeds in a sunflower.
For example, although there are thousands of kinds of flowers, there are relatively few consistent sets of numbers of petals on flowers. Some flowers have 3 petals; others have 5 petals; still others have 8 petals; and others have 13, 21, 34, 55, or 89 petals. There are exceptions and variations in these patterns, but they are comparatively few. All of these numbers observed in the flower petals—3, 5, 8, 13, 21, 34, 55, 89—appear in the Fibonacci series.
Similarly, the configurations of seeds in a giant sunflower and the configuration of rigid, spiny scales in pine cones also conform with the Fibonacci series. The corkscrew spirals of seeds that radiate outward from the center of a sunflower are most often 34 and 55 rows of seeds in opposite directions, or 55 and 89 rows of seeds in opposite directions, or even 89 and 144 rows of seeds in opposite directions. The number of rows of the scales in the spirals that radiate upwards in opposite directions from the base in a pine cone are almost always the lower numbers in the Fibonacci sequence—3, 5, and 8.
Why are Fibonacci numbers in plant growth so common? One clue appears in Fibonacci's original ideas about the rate of increase in rabbit populations. Given his time frame and growth cycle, Fibonacci's sequence represented the most efficient rate of breeding that the rabbits could have if other conditions were ideal. The same conditions may also apply to the propagation of seeds or petals in flowers. That is, these phenomena may be an expression of nature's efficiency. As each row of seeds in a sunflower or each row of scales in a pine cone grows radially away from the center, it tries to grow the maximum number of seeds (or scales) in the smallest space. The Fibonacci sequence may simply express the most efficient packing of the seeds (or scales) in the space available.
Gies, Joseph, and Frances Gies. Leonardo of Pisa and the New Mathematics of the Middle Ages. New York: Thomas Y. Crowell Co., 1969.
Swetz, Frank J. Capitalism & Arithmetic: The New Math of the 15th Century. LaSalle, Illinois: Open Court Press, 1987.
Stewart, Ian. "Mathematical Recreations: Daisy, Daisy, Give Me Your Answer, Do." Scientific American 272.1 (January 1995): 96-99.
Stewart, Ian. "Mathematical Recreations: Fibonacci Forgeries." Scientific American 272.5 (May 1995): 102-105.