A 1202 book that also brought Hindu-Arabic numerals to Europe, containing a rabbit-breeding puzzle (1202)
Liber Abaci by Leonardo of Pisa; the rabbit problem is a small part of a book mainly about arithmetic and commerce
Science · Mathematics
The Fibonacci sequence
1202
Mathematics — number sequences
The sequence 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89 …, in which each term is the sum of the two before it. It reached Europe in the Liber Abaci of 1202, in which Leonardo of Pisa — known since the nineteenth century as Fibonacci — posed a puzzle about breeding rabbits, though the sequence had been described centuries earlier by Indian scholars of Sanskrit prosody such as Pingala, Virahanka and Hemachandra. The ratio of consecutive terms tends to the golden ratio, and the numbers turn up in the spiral counts of pine cones, pineapples and sunflower heads.
How to spot them
- ▸0, 1, 1, 2, 3, 5, 8, 13, 21, 34 — each term the sum of the previous two
- ▸Introduced to Europe in a 1202 book by Leonardo of Pisa, in a problem about breeding rabbits
- ▸Ratios of consecutive terms converge on roughly 1.618
- ▸The counts of spirals in a pine cone or sunflower are almost always consecutive terms
- ▸Known in India centuries earlier from the study of Sanskrit metre
Key facts
The counts of clockwise and anticlockwise spirals in a sunflower head, usually 34 and 55
Phyllotaxis — consecutive terms of the sequence appear because of the golden angle
Binet's closed formula, which gives the nth term using powers of (1+√5)/2 (1843)
Named after Jacques Philippe Marie Binet, though de Moivre and Euler had it earlier
Each term is the sum of the two before it, starting 0, 1, 1, 2, 3, 5, 8
The defining recurrence, the simplest non-trivial linear recurrence relation
A quarterly journal founded in 1963 devoted entirely to one sequence and its relatives (1963)
The Fibonacci Quarterly, published by the Fibonacci Association
The number of ways to climb a staircase taking one or two steps at a time
A standard combinatorial appearance, equivalent to the original recurrence
The tiling of a plane by squares of side 1, 1, 2, 3, 5, 8 … spiralling outwards
The whirling-squares construction whose quarter-circle arcs give the familiar spiral
Key ideas
- The recurrence F(n) = F(n−1) + F(n−2) is the archetype of a linear recurrence; the ratio of successive terms converges to the positive root of x² = x + 1.
- Plants produce these numbers not by counting but because new growth at a fixed angle of about 137.5° packs most efficiently, and that angle is tied to the same ratio.
Quick-fire facts
- •Leonardo of Pisa (c. 1170 – c. 1250) published the Liber Abaci in 1202; the nickname 'Fibonacci' — 'son of Bonaccio' — was coined by the historian Guillaume Libri in 1838
- •The sequence appears in Indian prosody from Pingala onwards, and explicitly in Virahanka (c. 700) and Hemachandra (1150)
- •The rabbit problem asks how many pairs are produced in a year from one pair, assuming each pair breeds from its second month
- •The ratio of consecutive terms tends to the golden ratio; Kepler noted this in 1611
- •Every third term is even, every fourth is a multiple of 3, and the greatest common divisor of two terms is the term at the index equal to their indices' gcd
- •Zeckendorf's theorem states that every positive integer is a unique sum of non-consecutive terms of the sequence
- •Consecutive terms give convenient conversions between miles and kilometres: 5 miles is about 8 km, 8 is about 13
- •The Fibonacci Quarterly has been published since 1963 by the Fibonacci Association
- •The related Lucas numbers, 2, 1, 3, 4, 7, 11, 18 …, obey the same rule from different starting values
- •Only 0, 1, 8 and 144 are terms of the sequence that are also perfect powers, proved by Bugeaud, Mignotte and Siksek in 2006
Drill it — 8 cards
1 / 8new card
Sources
- MacTutor History of Mathematics Archive, University of St Andrews — Leonardo Pisano Fibonacci
- Keith Devlin — The Man of Numbers: Fibonacci's Arithmetic Revolution