Discover the Fibonacci Sequence & the Golden Ratio

Learn More

What is Fibonacci?

Fibonacci refers to the sequence of numbers made famous by thirteenth-century mathematician Leonardo Pisano, who presented and explained the solution to an algebraic math problem in his book Liber Abaci (1228). The Fibonacci sequence and the ratios of its sequential numbers have been discovered to be pervasive throughout nature, art, music, biology, and other disciplines. The sequence begins with 0 and 1 and is comprised of subsequent numbers in which the nth number is the sum of the two previous numbers. The equation for finding a Fibonacci number can be written like this:

Fn = F(n-1) + F(n-2). The starting points are F1 = 1 and F2 = 1.

Each number in the Fibonacci sequence is identified with a subscript 1, 2, 3, 4 …… to indicate which term of the sequence we are talking about. Thus F16 refers to the sixteenth Fibonacci number.

The Fibonacci Sequence

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987 . . .

What is the Golden Ratio?

Related to the Fibonacci sequence is another famous mathematic term: the Golden Ratio. 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). 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. 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. In the image below, the ratio of the smaller part of a line (CB), to the larger part (AC) – i.e. CB/AC – is the same as the ratio of the larger part, AC, to the whole line AB. Therefore, CB/AC = AC/AB. Golden Section Phi and phi are also known as the Golden Number and the Golden Section. 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.

Leonardo Pisano

Who is Leonardo Pisano (Fibonacci)?

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. 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. This remarkable endorsement did not resuscitate Leonardo’s legacy, however, and his name was once more quickly forgotten.

For 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. This was in 1797, over five centuries after Leonardo had died. Remarkably,  it was yet another hundred years before Leonardo would once again be acknowledged academically and given the credit to which he is due.

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). Leonardo has been called ‘Fibonacci’ ever since. READ MORE

Fibonacci in Art & Music

Like Euclid before him, ‘Fibonacci’ believed that calculation was an art form; to him, it was a marvelous thing of beauty pervasive in art and sound.

Learn More


Leonardo Pisano’s seminal work, Liber Abaci, was the fountainhead of mathematical advances in Europe in the Middle Ages and influential in replacing Roman numerals with modern Arabic numerals.

Learn More


Scientific research finds evidence that the Fibonacci numbers and the Golden Ratio are prevalent in natural objects, from the microscopic structure proportions in the bodies of living beings on Earth to the relationships of gravitational forces and distances between bodies in the universe.

Learn More


In increasing numbers, serious mathematics scholars today are beginning to recognize that Leonardo Pisano may have been the “most noteworthy mathematical genius of the Middle Ages,” and definitely the most influential of all medieval writers in promoting the Hindu-Arabic numerals to European scholars.

Learn More


[mc4wp_form id=”71″]