What is the Fibonacci Sequence?

The shapes of spiral galaxies, such as Messier 74, and hurricanes, such as Hurricane Irene, follow the Fibonacci sequence.

The shapes of spiral galaxies, such as Messier 74, and hurricanes, such as Hurricane Irene, follow the Fibonacci sequence.

The Fibonacci sequence is a series of numbers where a number is found by adding up the two numbers before it. Starting with 0 and 1, the sequence goes 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, and so forth. Written as a rule, the expression is xn = xn-1 + xn-2.

Named after Fibonacci, also known as Leonardo of Pisa or Leonardo Pisano, Fibonacci numbers were first introduced in his Liber abaci in 1202. The son of a Pisan merchant, Fibonacci traveled widely and traded extensively. Math was incredibly important to those in the trading industry, and his passion for numbers was cultivated in his youth.

Knowledge of numbers is said to have first originated in the Hindu-Arabic arithmetic system, which Fibonacci studied while growing up in North Africa. Prior to the publication of Liber abaci, the Latin-speaking world had yet to be introduced to the decimal number system. He wrote many books about geometry, commercial arithmetic and irrational numbers. He also helped develop the concept of zero.

Fibonacci first noted the sequence when pondering a mathematical problem about rabbit breeding. Beginning with a male and female rabbit, how many pairs of rabbits could be born in a year? The problem assumes the following conditions:

  • Begin with one male rabbit and female rabbit that have just been born.
  • Rabbits reach sexual maturity after one month.
  • The gestation period of a rabbit is one month.
  • After reaching sexual maturity, female rabbits give birth every month.
  • A female rabbit gives birth to one male rabbit and one female rabbit.
  • Rabbits do not die.

This is best understood in this diagram:

After one month, the first pair is not yet at sexual maturity and can’t mate. At two months, the rabbits have mated but not yet given birth, resulting in only one pair of rabbits. After three months, the first pair will give birth to another pair, resulting in two pairs. At the fourth month mark, the original pair gives birth again, and the second pair mates but does not yet give birth, leaving the total at three pair. This continues until a year has passed, in which there will be 233 pairs of rabbits.

Sunflower seeds grow in Fibonacci spirals.
                     Sunflower seeds grow in Fibonacci spirals.

Though the rabbit question may pose completely unrealistic conditions, Fibonacci numbers do actually appear in nature, from sunflowers to hurricanes to galaxies. Sunflowers seeds, for example, are arranged in a Fibonacci spiral, keeping the seeds uniformly distributed no matter how large the seed head may be.

A Fibonacci spiral is a series of connected quarter-circles drawn inside an array of squares with Fibonacci numbers for dimensions. The squares fit perfectly together because of the nature of the sequence, where the next number is equal to the sum of the two before it. Any two successive Fibonacci numbers have a ratio very close to the Golden Ratio, which is roughly 1.618034. The larger the pair of Fibonacci numbers, the closer the approximation. The spiral and resulting rectangle are known as the Golden Rectangle.

The Golden Ratio is denoted by the Greek letter phi. Greek architects used the ratio 1:phi as an integral part of their designs, including the Parthenon in Athens. Though this was not consciously used by Greeks or artists, the Golden Rectangle does appear in the Mona Lisa and other Renaissance art works. Phi is also the ratio of the side of a regular pentagon to its diagonal. The resulting pentagram forms a star, which is the star seen on many flags.

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