Sacred geometry · The numbers
What is the Fibonacci sequence?
The Fibonacci sequence is a list of numbers where each one is the sum of the two before it: 1, 1, 2, 3, 5, 8, 13, 21, 34, and on forever. Divide any number in the list by the one before it and the answer gets closer and closer to 1.618, the golden ratio. Leonardo of Pisa made it famous in Europe with a puzzle about rabbits in 1202, but poets in India had counted the same numbers centuries earlier.
Last number
144
Last ÷ the one before
The rule
The first 25 Fibonacci numbers
Start with 1 and 1. Add them to get the next number. Keep adding the last two. That is the whole rule. Mathematicians write it as F(n) = F(n−1) + F(n−2), and many start the list with a 0 in front, so F(0) = 0, F(1) = 1, F(2) = 1.
The numbers grow fast. By the 25th you are at 75,025; by the 60th, past a trillion. Every third number is even (2, 8, 34, 144, 610...), because odd plus odd makes even and the pattern odd, odd, even repeats forever.
Try the second view in the machine above. Start with any two numbers you like, say 7 and 100. Keep adding the last two. Within about 20 steps, the last number divided by the one before is 1.618 to many decimal places. The starting numbers do not matter. The adding does.
1202
The rabbit puzzle
Leonardo of Pisa, born around 1170, grew up partly in North Africa, where his father represented Pisa’s merchants. He came home and wrote Liber Abaci, the Book of Calculation, in 1202. Its main job was to bring the Hindu-Arabic numerals 0 to 9 into Europe. Near the end of chapter 12 sits a word problem:
“A certain man had one pair of rabbits together in a certain enclosed place, and one wishes to know how many are created from the pair in one year when it is the nature of them in a single month to bear another pair, and in the second month those born to bear also.”Liber Abaci, 1202 · translated by L. E. Sigler, 2002
Count the pairs month by month and you get 1, 2, 3, 5, 8, 13... Leonardo worked it through to 377 pairs at the end of the year. The rabbits are not realistic. The point was the counting, and the counting is the sequence.
He never called himself Fibonacci. The nickname, short for filius Bonacci, “son of Bonacci,” was attached to him long after his death; it shows up in print from the late 1700s. The French mathematician Édouard Lucas gave the sequence the name Fibonacci in the 1870s.
Before Fibonacci
Indian poets counted it first
In Sanskrit poetry, a syllable is either short (one beat) or long (two beats). How many rhythms fill a line of a given length? One beat: 1 way. Two beats: 2 ways. Three beats: 3 ways. Four beats: 5 ways. Each answer is the sum of the two before, because a rhythm ends with either a short or a long syllable.
The historian Parmanand Singh traced this in a 1985 paper. The grammarian Pingala, writing on poetic metre roughly between 450 and 200 BCE, is the first whose work shows knowledge of the numbers. Virahanka gave them between 600 and 800 CE, Gopala before 1135, and the Jain scholar Hemachandra around 1150, still before Leonardo’s book. Hemachandra simply listed them: “1, 2, 3, 5, 8, 13, 21, 34 and in this way, afterwards.”
The formula
Jump straight to any Fibonacci number
You do not have to add your way up. There is a direct formula, known as Binet’s formula after Jacques Binet, who published it in 1843. Abraham de Moivre had found it in 1718.
F(n) = (φⁿ − ψⁿ) ÷ √5
Here φ is 1.618... and ψ is −0.618... Because ψⁿ shrinks to almost nothing, there is a shortcut: raise 1.618 to the power n, divide by √5 (2.236), and round. For n = 10: 1.618¹⁰ ÷ 2.236 = 55.004, which rounds to 55, the 10th Fibonacci number.
Why the ratio goes to 1.618. When the numbers are big, each new number is the old one plus the one before. So the ratio r has to satisfy r = 1 + 1/r. The only positive number that does that is the golden ratio. Kepler noted the ratio in a letter in 1609; Robert Simson is usually credited with the proof in 1753.
Out in the world
Sunflowers, pine cones and pineapples
Look at the seeds in a sunflower head and you see two sets of spirals, one turning each way. Count them and you very often get neighbouring Fibonacci numbers, such as 34 and 55, or 55 and 89. Pineapples usually show 8 and 13 rows of eyes. Pine cones studied in California showed 5, 8 or 13 spirals.
The reason is the golden angle, about 137.5°. A plant that puts each new seed or leaf at that turn from the last one packs them with no gaps, and the spirals that appear come in Fibonacci numbers. In 2012 a Manchester museum asked the public to grow sunflowers for Alan Turing’s centenary. Of 768 spiral counts in the best data, 565, about 74%, were Fibonacci numbers, and 82% had Fibonacci structure. The rest did not, which is why the honest word is “often,” not “always.”
The same numbers turn up in pure arithmetic too. Take the digital root of every Fibonacci number and the pattern repeats every 24 steps; the Fibonacci Clock shows that wheel for any number.
What is proven
The Fibonacci sequence, sorted
| Claim | Status |
|---|---|
| Each number is the sum of the two before it. | Proven · the definition |
| The ratio of neighbours heads to 1.618. | Proven · Kepler 1609, Simson 1753 |
| Any two starting numbers head to 1.618. | Proven · try it above |
| Leonardo of Pisa wrote the rabbit puzzle in 1202. | Real · Liber Abaci, chapter 12 |
| Fibonacci discovered the sequence. | No · Indian poets knew it centuries earlier |
| Sunflowers always show Fibonacci spirals. | Often · 74% to 82% in a 2016 study |
Questions
The Fibonacci sequence
What are the first 10 Fibonacci numbers?
1, 1, 2, 3, 5, 8, 13, 21, 34, 55. If you start the list at zero, as many textbooks do, they are 0, 1, 1, 2, 3, 5, 8, 13, 21, 34.
How do you explain the Fibonacci sequence simply?
Start with 1 and 1, and keep adding the last two numbers to get the next one: 1 + 1 = 2, 1 + 2 = 3, 2 + 3 = 5, 3 + 5 = 8, and so on forever.
Why is the Fibonacci sequence so important?
It is one of the simplest rules that produces the golden ratio, 1.618, and it turns up in plant spirals, in counting problems and in computer science. It also gave its name to a whole field: The Fibonacci Quarterly has published research on it since 1963.
What is the formula for the Fibonacci sequence?
F(n) = F(n−1) + F(n−2), starting from 1 and 1. To jump straight to any term, use Binet’s formula: F(n) = (φⁿ − ψⁿ) ÷ √5, where φ = 1.618... and ψ = −0.618..., or simply round φⁿ ÷ √5.
Is the Fibonacci sequence in the Bible?
No. It is first recorded in Indian works on poetic metre, from Pingala onward, and reached Europe through Leonardo of Pisa’s book Liber Abaci in 1202.
Did Leonardo da Vinci use the Fibonacci sequence?
There is no record of it. Leonardo da Vinci’s link is to the golden ratio: he drew the figures for Luca Pacioli’s book Divina proportione, printed in 1509.
Sources
Where this comes from
- Eric W. Weisstein, “Fibonacci Number,” MathWorld: mathworld.wolfram.com.
- J. J. O’Connor and E. F. Robertson, “Leonardo Pisano Fibonacci,” MacTutor, University of St Andrews: st-andrews.ac.uk.
- Keith Devlin, “Recreational mathematics in Leonardo of Pisa’s Liber abbaci,” Stanford University, quoting L. E. Sigler’s 2002 translation: stanford.edu.
- Parmanand Singh, “The So-called Fibonacci Numbers in Ancient and Medieval India,” Historia Mathematica 12 (1985): doi.org.
- Jeff Miller, “Earliest Known Uses of Some of the Words of Mathematics (F),” MacTutor: st-andrews.ac.uk.
- “Jacques Binet,” MacTutor, on Binet 1843 and de Moivre 1718: st-andrews.ac.uk.
- J. J. O’Connor and E. F. Robertson, “The Golden ratio,” MacTutor, on Kepler 1609 and Simson 1753: st-andrews.ac.uk.
- Eric W. Weisstein, “Lucas Number” and “Cassini’s Identity,” MathWorld: mathworld.wolfram.com.
- Eric W. Weisstein, “Phyllotaxis,” MathWorld, on 34 and 55 sunflower spirals: mathworld.wolfram.com.
- P. B. Onderdonk, “Pineapples and Fibonacci Numbers,” The Fibonacci Quarterly 8 (1970): fq.math.ca.
- R. Knott, Fibonacci Numbers and Nature, University of Surrey, summarising Brousseau 1968 on pine cones: surrey.ac.uk.
- J. Swinton, E. Ochu and the MSI Turing’s Sunflower Consortium, “Novel Fibonacci and non-Fibonacci structure in the sunflower,” Royal Society Open Science 3 (2016): doi.org.
- The Fibonacci Association and The Fibonacci Quarterly, since 1963: fibonacciassociation.org.
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An image of the squares and spiral. Copy link reopens your exact numbers.
Poster
The Fibonacci squares and spiral, printed large.