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Fibonacci Sequence

Fibonacci Sequence


Other Fibonacci sequences

The Fibonacci sequence in nature


The Fibonacci sequence is a series of numbers in which each succeeding number (after the second) is the sum of the previous two. The most famous Fibonacci sequence is 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89. . .. This sequence expresses many naturally occurring relationships in the plant world.


The Fibonacci sequence was invented by the Italian Leonardo Pisano Bigollo (11801250), who is known in mathematical history by several names: Leonardo of Pisa (Pisano means from Pisa) and Fibonacci (which means son of Bonacci). Fibonacci, the son of an Italian businessman from the city of Pisa, grew up in a trading colony in North Africa during the Middle Ages. Italians were some of the western worlds most proficient traders and merchants during the Middle Ages, and they needed arithmetic to keep track of their commercial transactions. Mathematical calculations were made using the Roman numeral system (I, II, III, IV, V, VI, etc.), but that system made it hard to do the addition, subtraction, multiplication, and division that merchants needed to keep track of their transactions.

While growing up in North Africa, Fibonacci learned the more efficient Hindu-Arabic system of arithmetical notation (1, 2, 3, 4. . .) from an Arab teacher. In 1202, he published his knowledge in a famous book called the Liber Abaci (which means the book of the abacus, even though it had nothing to do with the abacus). The Liber Abaci showed how superior the Hindu-Arabic arithmetic system was to the Roman numeral system, and it showed how the Hindu-Arabic system of arithmetic could be applied to benefit Italian merchants.

The Fibonacci sequence was the outcome of a mathematical problem about rabbit breeding that was posed in the Liber Abaci. The problem was this: Beginning with a single pair of rabbits (one male and one female), how many pairs of rabbits will be born in a year, assuming that every month each male and female rabbit gives birth to a new pair of rabbits, and the new pair of rabbits itself starts giving birth to additional pairs of rabbits after the first month of their birth?

Table 1 illustrates one way of looking at Fibonaccis solution to this problem. Each number in the table represents a pair of rabbits. Each pair of rabbits can only give birth after its first month of life. Beginning in the third month, the number in the Mature Pairs column represents the number of pairs that can bear rabbits. The numbers in the Total Pairs column represent the Fibonacci sequence.

Other Fibonacci sequences

Although the most famous Fibonacci sequence is 1, 1, 2, 3, 5, 8, 13, 21, 34, 55. . ., a Fibonacci sequence may be any series of numbers in which each succeeding

Table 1. (Thomson Gale.)
Estimation using Fibonacci sequence
 Newborns (cant reproduce) One-month-olds (cant reproduce) Mature (can reproduce) Total pairs
Month 11+0+0=1
Month 20+1+0=1
Month 31+0+1=2
Month 41+1+1=3
Month 52+1+2=5
Month 63+2+3=8
Month 75+3+5=13
Month 88+5+8=21
Month 913+8+13=34
Month 1021+13+21=55

number (after the second) is the sum of the previous two. That means that the specific numbers in a Fibonacci series depend upon the initial numbers. Thus, if a series begins with 3, then the subsequent series would be as follows: 3, 3, 6, 9, 15, 24, 39, 63, 102, and so on.

A Fibonacci series can also be based on something other than an integer (a whole number). For example, the series 0.1, 0.1, 0.2, 0.3, 0.5, 0.8, 1.3, 2.1, 3.4, 5.5, and so on, is also a Fibonacci sequence.

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 petals3, 5, 8, 13, 21, 34, 55, 89appear 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


Phyllotaxis The arrangement of the leaves of a plant on a stem or axis.

Radially Diverging outward from a center, as spokes do from a wagon wheel or as light does from the sun.

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 sequence3, 5, and 8.

Why are Fibonacci numbers in plant growth so common? One clue appears in Fibonaccis original ideas about the rate of increase in rabbit populations. Given his time frame and growth cycle, Fibonaccis 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 natures 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.

See also Integers; Numeration systems.



Cummins, Steven. Fibonacci Sequence. Philadelphia, PA: Xlibris Corporation, 2005.

Posamentier, Alfred S. and Ingmar Lehmann. The (Fabulous) Fibonacci Numbers. Amherst, NY: Prometheus Books, 2007.

Shesso, Renna. Math for Mystics: Fibonacci Sequence, Lunas Labyrinth, Golden Sections and Other Secrets. Red Wheel Weiser, 2007.

Patrick Moore

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