Fibonacci Sequence

Fibonacci Sequence

The Fibonacci Sequence is a series of numbers in which each number ( Fibonacci number ) is the sum of the two preceding numbers, for example, the series 1, 1, 2, 3, 5, 8, etc., is a Fibonacci Sequence. The sequence was named after a man called Leonardo of Pisa, who was nicknamed Fibonacci.

However, Fibonacci did not discover Fibonacci numbers. He learnt about them during his travels to North Africa, and he introduced them to Europe. Fibonacci numbers were used in ancient India much before they were used in Europe. Indians used it for the metrical sciences, also known as prosody (the study of a poetic meter).

Prosody was important in ancient Indian ritual because of an emphasis on the purity of utterance. Virahanka gave the method for the formation of the Fibonacci sequence (between A.D. 600 and 800), Gopala (before A.D. 1135), and Hemacandra (c. A.D. 1150), all prior to Leonardo Fibonacci (c. A.D. 1202).

In fact, Fibonacci himself wrote that he had studied Indian numbers and did not come up with the number series. In this sequence, each number is the sum of the previous two numbers, so it begins 0, 1, 1, 2, 3, 5, 8, 13, 21, 34 and so on. If you take two consecutive numbers, their ratio is very close to 1.618. This isn’t that impressive on its own until you realise that this ratio is the key to everything from encrypting computer data, to the numbers of spirals on a sunflower head, our limbs and why the Mona Lisa is so pleasing to the eye.

Sunflower 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 numbers before it.

In the 1750s, Robert Simson noted that the ratio of each term in the Fibonacci Sequence to the previous term approaches, with ever greater accuracy the higher the terms, a ratio of approximately 1:1.6180339887. This value is referred to as the Golden Ratio, also known as the Golden Mean, Golden Section, Divine Proportion, etc. and is usually denoted by the Greek letter phi φ (or sometimes the capital letter Phi Φ).

Essentially, two quantities are in the Golden Ratio if the ratio of the sum of the quantities to the larger quantity is equal to the ratio of the larger quantity to the smaller one.

Fun Fact

Fibonacci numbers are used to estimate the price of a stock in the future based on certain ratios found within the Fibonacci numbers.