![]() So plants that tend toward this value have an advantage against plants that don't. It turns out that if a plant grows one leaf, then the next phi (the golden ratio) rotations from the first, then the third phi rotations from the second, and the fourth phi rotations from the third, and so on, that process will result in the longest possible time before the newest leaf is in the shadow of any existing leaf. same problem.Īs this species evolves, the plants whose leaves are most often useful have an advantage and breed more. If the second leaf is opposite the first then that is good, but the third will be in shadow and useless. If it grows a second leaf in the shadow of the first then that leaf is useless. ![]() And there are places in the natural world were extreme irrationality is the most efficient solution to a problem, so by natural selection living systems tend toward that value where it works best.Ĭonsider a plant that has grown one leaf. ![]() Leonardo has been called ‘Fibonacci’ ever since.It turns out that the golden ratio is not only an irrational number. 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. 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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