Applications of Fibonacci numbers include computer algorithms such as the Fibonacci search technique and the Fibonacci heap data structure, and graphs called Fibonacci cubes used for interconnecting parallel and distributed systems. įibonacci numbers appear unexpectedly often in mathematics, so much so that there is an entire journal dedicated to their study, the Fibonacci Quarterly. They are named after the Italian mathematician Leonardo of Pisa, also known as Fibonacci, who introduced the sequence to Western European mathematics in his 1202 book Liber Abaci. The Fibonacci numbers were first described in Indian mathematics, as early as 200 BC in work by Pingala on enumerating possible patterns of Sanskrit poetry formed from syllables of two lengths. The sequence commonly starts from 0 and 1, although some authors start the sequence from 1 and 1 or sometimes (as did Fibonacci) from 1 and 2. Numbers that are part of the Fibonacci sequence are known as Fibonacci numbers, commonly denoted F n. In mathematics, the Fibonacci sequence is a sequence in which each number is the sum of the two preceding ones. Leonardo has been called ‘Fibonacci’ ever since.A tiling with squares whose side lengths are successive Fibonacci numbers: 1, 1, 2, 3, 5, 8, 13 and 21 In the 1870s, the French mathematician Edouard Lucas assigned the name “Fibonacci” to the number sequence that is the solution to the famous “Rabbit Problem” in Leonardo Pisano’s book, Liber Abaci (1228). Remarkably, it was yet another hundred years before Leonardo would once again be acknowledged academically and given the credit to which he is due. This was in 1797, over five centuries after Leonardo had died. This remarkable endorsement did not resuscitate Leonardo’s legacy, however, and his name was once more quickly forgotten.įor another three hundred years historical anonymity obscured the achievements of Leonardo Pisano until one day, by slim chance, a mathematics historian named Pietro Cossali (1748-1815) noticed Pacioli’s reference and began researching Leonardo’s works on his own. No biographies were written about him or his many accomplishments in math even mathematicians did not know who he was until 1494, when a respected Italian mathematician named Luca Pacioli (1447-1517) briefly mentioned Leonardo’s name in the introduction to a book of his own, Summa, giving credit to him for most of the ideas presented in his own book. Master Leonardo Pisano (not to be confused with Leonardo da Vinci) was a beloved public servant of Pisa, Italy, who achieved fame during his lifetime (ca.1170 – ca.1250) but was forgotten within two hundred years.
The formula for Golden Ratio is: F(n) = (x^n – (1-x)^n)/(x – (1-x)) where x = (1+sqrt 5)/2 ~ 1.618 The Golden Ratio represents a fundamental mathematical structure which appears prevalent – some say ubiquitous – throughout Nature, especially in organisms in the botanical and zoological kingdoms.
Phi and phi are also known as the Golden Number and the Golden Section. CB/AC – is the same as the ratio of the larger part, AC, to the whole line AB. In the image below, the ratio of the smaller part of a line (CB), to the larger part (AC) – i.e. Phi (Φ), 1.61803 39887…, is also the number derived when you divide a line in mean and extreme ratio, then divide the whole line by the largest mean section its inverse is phi (φ), 0.61803 39887…, obtained when dividing the extreme (smaller) portion of a line by the (larger) mean. After these first ten ratios, the quotients draw ever closer to Phi and appear to converge upon it, but never quite reach it because it is an irrational number.
#FIBONACCI SEQUENCE FORMULA GOLDEN RATIO SERIES#
When a number in the Fibonacci series is divided by the number preceding it, the quotients themselves become a series that follows a fascinating pattern: 1/1 = 1, 2/1 = 2, 3/2 = 1.5, 5/3 = 1.666…, 8/5 = 1.6, 13/8 = 1.625, 21/13 = 1.61538, 34/21 = 1.619, 55/34 = 1.6176…, and 89/55 = 1.618… The first ten ratios approach the numerical value 1.618034… which is called the “Golden Ratio” or the “Golden Number,” represented by the Greek letter Phi (Φ, φ). Related to the Fibonacci sequence is another famous mathematic term: the Golden Ratio.
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