The Fibonacci Sequence and the Golden Ratio Explained
Each Fibonacci number is the sum of the previous two, and their ratio approaches the golden ratio, about 1.618. See the math, the history, and the myths.
The Fibonacci sequence is the list of numbers 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, and so on, in which each number is the sum of the two before it. The ratio of neighbouring terms approaches the golden ratio, about 1.618, usually written with the Greek letter phi. The sequence turns up in counting problems, computer algorithms, and the spiral arrangement of some plants, but many popular claims about it, especially about art and the human body, are exaggerated.
What is the Fibonacci sequence?
The rule is simple: start with 0 and 1, then add the last two numbers to get the next. The sequence is named after Leonardo of Pisa, known as Fibonacci, who included it in his 1202 book Liber Abaci in a problem about breeding rabbits: if each pair produces a new pair every month from its second month on, how many pairs are there after a year? The answer is 377. The numbers were not new even then. Indian mathematicians had described them centuries earlier, in counting the ways to arrange short and long syllables in Sanskrit poetry.
How does it connect to the golden ratio?
The golden ratio is the number (1 + the square root of 5) divided by 2, about 1.6180339887. It is the positive solution of the equation x squared equals x plus 1, and it describes a division of a line in which the whole is to the larger part as the larger part is to the smaller. If you divide each Fibonacci number by the one before it, the results settle toward this value, alternately a little above and a little below:
| Term | Value | Ratio to the previous term |
|---|---|---|
| 5th | 5 | 1.667 |
| 6th | 8 | 1.600 |
| 7th | 13 | 1.625 |
| 8th | 21 | 1.615 |
| 9th | 34 | 1.619 |
| 10th | 55 | 1.618 |
The link is exact, as Binet's formula shows:
F(n) = (phi^n − psi^n) / sqrt(5)
phi = (1 + sqrt(5)) / 2 psi = (1 − sqrt(5)) / 2
Because psi is a number between −1 and 0, its powers shrink rapidly, so the nth Fibonacci number is simply the whole number closest to phi to the nth power divided by the square root of 5.
Where does it appear in mathematics and computing?
The numbers count many things. The number of ways to tile a strip two units wide and n units long with dominoes is a Fibonacci number, and Zeckendorf's theorem says every positive whole number can be written in exactly one way as a sum of non-consecutive Fibonacci numbers. In computer science, the naive recursive way of computing Fibonacci numbers takes exponential time because it repeats the same calculations, which makes it the standard teaching example for dynamic programming and for why Big O notation matters. Euclid's algorithm for the greatest common divisor is slowest when given two consecutive Fibonacci numbers, a result known as Lamé's theorem from 1844.
Does the Fibonacci sequence appear in nature?
Sometimes, and for a concrete reason. Many flowers have petal counts from the sequence: lilies have 3, buttercups 5, some delphiniums 8, marigolds 13, and asters 21. The best-known case is the arrangement of seeds in a sunflower head, which forms two sets of spirals, one clockwise and one counterclockwise, whose counts are often neighbouring Fibonacci numbers such as 34 and 55, or 55 and 89. The cause is the golden angle, about 137.5 degrees, which is the angle you get by dividing a full turn in the golden ratio. If each new seed or leaf appears a golden angle around from the last, the elements pack evenly and never line up in rows. For leaves this reduces shading and improves light capture for photosynthesis, the kind of efficiency that natural selection would favour. Many plants do not follow the pattern, however, and counts vary.
What are the myths about the golden ratio?
Popular claims that the Parthenon, the Mona Lisa, and the proportions of the human face are built on the golden ratio fall apart on inspection. Measurements of the Parthenon can be made to fit many ratios depending on what is measured, and there is no evidence that Greek architects used the golden ratio on purpose. Luca Pacioli published an admiring book, Divina Proportione, in 1509 with illustrations by Leonardo da Vinci, but nothing shows that Leonardo used it in the Mona Lisa. The shell of the nautilus is a logarithmic spiral, but its growth rate is not the golden ratio. In finance, the so-called Fibonacci retracement levels used by traders have little reliable evidence behind them.
The real golden ratio is plenty impressive on its own. It is the number that is hardest to approximate by fractions, which is why it appears in the way some plants and some mathematical systems avoid repeating themselves.
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fibonacci golden ratio mathematics number theory patterns in nature
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