What is the Fibonacci sequence? Patterns in numbers

The Fibonacci sequence is the list of numbers 0, 1, 1, 2, 3, 5, 8, 13, 21, 34 and so on, in which each number is the sum of the two before it. It is named after Leonardo of Pisa, known as Fibonacci, who used it in a puzzle about rabbits in 1202. The ratio between neighboring numbers settles toward the golden ratio, about 1.618, and the numbers turn up in the spirals of some plants.

In short

• One rule makes the whole sequence: add the last two numbers to get the next one.
• Divide any Fibonacci number by the one before it and the answer closes in on the golden ratio, 1.6180339…
• Sunflower and pine-cone spirals often come in Fibonacci numbers, but many claims about the golden ratio in buildings, art and faces don't hold up.

How is the sequence built?

Start with 0 and 1. Add them to get 1. Add the last two, 1 and 1, to get 2. Then 1 + 2 = 3, 2 + 3 = 5, 3 + 5 = 8, and so on without end. Written as a rule, F(n) = F(n − 1) + F(n − 2), with F(0) = 0 and F(1) = 1. Some books start at 1, 1 instead of 0, 1; the rest of the sequence is the same.

The numbers grow fast. The 10th is 55, the 20th is 6,765, and the 30th is 832,040. Each one is roughly 1.618 times the one before, so the sequence grows almost exactly like repeated multiplication by that number. A formula known to Abraham de Moivre in the 18th century, and later named after the French mathematician Jacques Binet, even skips the adding: F(n) is the golden ratio raised to the power n, divided by the square root of 5, rounded to the nearest whole number. For n = 10, that is 122.99 ÷ 2.236 = 55.004, which rounds to 55.

Where did the Fibonacci sequence come from?

Leonardo of Pisa, born around 1170, grew up partly in North Africa, where his father worked as a trade official, and learned the numerals used by Indian and Arab mathematicians. His book Liber abaci (1202, revised in 1228) taught Europe to calculate with those nine digits and the zero instead of Roman numerals, according to the MacTutor history of mathematics at the University of St Andrews. The rabbit puzzle was one exercise among hundreds.

It went like this. A man puts a pair of rabbits in a walled place. Each month, every pair old enough to breed has a new pair, and a new pair can breed from its second month. No rabbit dies. How many pairs are there after a year? Count month by month and the totals run 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377. The answer in the book is 377 pairs. Fibonacci didn't claim real rabbits behave like this; it was an arithmetic exercise.

He also wasn't first. Indian scholars counting the rhythms of Sanskrit poetry had met the same numbers centuries earlier. If a line is built from short syllables worth one beat and long syllables worth two, the number of ways to fill a line of n beats is a Fibonacci number. Virahanka described this around 700 CE, and Hemachandra around 1150. The name "Fibonacci sequence" came much later, from the French mathematician Édouard Lucas in the 19th century.

How is it connected to the golden ratio?

The golden ratio, written φ (phi), is the number (1 + √5) ÷ 2, or 1.6180339… It is the one ratio with this property: if you split a line so that the whole is to the long part as the long part is to the short part, that ratio is φ. Divide each Fibonacci number by the one before it and the answers swing above and below φ, closer every time.

Ratio	Value	Distance from φ
5 ÷ 3	1.6667	+0.0486
8 ÷ 5	1.6000	−0.0180
13 ÷ 8	1.6250	+0.0070
21 ÷ 13	1.6154	−0.0026
34 ÷ 21	1.6190	+0.0010
55 ÷ 34	1.6176	−0.0004
89 ÷ 55	1.6182	+0.0001

The reason is the adding rule itself. If the ratio of neighbors settles at some value r, then the rule F(n) = F(n − 1) + F(n − 2) forces r = 1 + 1/r, and the only positive number that solves it is φ. Any sequence built by the same rule, starting from any two positive numbers, ends up at the same ratio. Like pi, φ is irrational: no fraction equals it exactly, and Fibonacci ratios are among the best fractions for approximating it.

Does the Fibonacci sequence really appear in nature?

Sometimes, and the best case is real. Look at the head of a sunflower or the base of a pine cone and you see two sets of spirals, one turning clockwise and one counterclockwise. Count them and you often get two neighboring Fibonacci numbers: 34 and 55, or 55 and 89, on sunflowers; 5 and 8, or 8 and 13, on pine cones. The cause is how the plant grows. New seeds or scales form at the center one after another, each turned about 137.5 degrees from the last, the "golden angle" that splits a circle in the golden ratio. That angle packs seeds without gaps, and the spirals that the eye picks out end up counted in Fibonacci numbers.

The pattern isn't universal, though. In 2012, a citizen-science project inspired by Alan Turing's last research counted spirals on 657 sunflowers grown by the public, and published the results in 2016. In the most reliable photos, 565 of 768 spiral counts were Fibonacci numbers, about 74%. Another 67 fit related patterns, and some heads had no Fibonacci structure at all.

Other appearances are counting facts rather than growth patterns. A male honeybee has only a mother, while a female has two parents, so a drone's ancestors in each generation back run 1, 2, 3, 5, 8, as Ron Knott's long-running page at the University of Surrey sets out.

Which golden ratio claims are myths?

The golden ratio attracts claims that fall apart when measured. A building or a face has dozens of ratios between 1 and 2, so with enough measuring, some will land near 1.618. Choosing the points afterward makes almost any shape look golden, which is a lesson in critical thinking as much as in geometry.

Claim	What the evidence shows
Sunflower and pine-cone spirals come in Fibonacci numbers	Mostly true, with real exceptions
A drone bee's ancestry follows the sequence	True, by how bees reproduce
The nautilus shell is a golden spiral	False: real shells are a flatter spiral than the golden one
The Parthenon was designed on the golden ratio	No evidence in Greek writing; measurements depend on which points are picked
Beautiful faces follow the golden ratio	Not supported; ratios can be redefined to fit
Leonardo da Vinci built the Vitruvian Man on it	His proportions don't match it, and he wrote in whole-number ratios

The mathematician Chris Budd sets out the evidence in Plus magazine, noting that the idea of "golden" architecture traces back only to the 1850s. The nautilus is the clearest case: the golden spiral has a growth constant of about 0.306, while the value most often measured in real shells is about 0.18.

What is the Fibonacci sequence used for?

Mostly inside mathematics and computing. Every whole number can be written in exactly one way as a sum of Fibonacci numbers with no two neighbors, a fact called Zeckendorf's theorem: 100 is 89 + 8 + 3. Computer scientists use Fibonacci numbers in a data structure called the Fibonacci heap, in a search method, and as a classic first exercise in recursion. Software teams that estimate work in "story points" often use 1, 2, 3, 5, 8, 13, because the widening gaps admit that big jobs are harder to size. It also shows how a rule simple enough for a child can carry structure that mathematicians are still studying, which is part of what science looks for in nature.

Check yourself

• Without looking back, write the first ten Fibonacci numbers and say the rule that makes them.
• Why do the ratios of neighboring Fibonacci numbers close in on 1.618, rather than some other number?
• What makes sunflower spirals a real Fibonacci pattern, and the Parthenon a doubtful one?

Sources

• Leonardo Pisano Fibonacci, biography — MacTutor History of Mathematics, University of St Andrews
• Novel Fibonacci and non-Fibonacci structure in the sunflower: results of a citizen science experiment — Swinton and Ochu, Royal Society Open Science, 2016
• The Fibonacci numbers and golden section in nature — Ron Knott, University of Surrey
• Myths of maths: The golden ratio — Chris Budd, Plus magazine, 2020
• Fibonacci sequence — Wikipedia