/ˌfiː.bəˈnɑː.tʃi ˈsiː.kwəns/

Fibonacci sequence

noun · number theory, composition

What it is

The Fibonacci sequence is defined by the recurrence F(n) = F(n−1) + F(n−2) with F(1) = F(2) = 1. The next twenty terms are 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765. The ratio F(n+1)/F(n) approaches φ from above and below in alternation — 2/1 = 2, 3/2 = 1.5, 5/3 ≈ 1.667, 8/5 = 1.6, 13/8 = 1.625, 21/13 ≈ 1.615, 34/21 ≈ 1.619 — converging to φ by the seventh term to four decimal places.

In composition, the practical use is the Fibonacci grid: a rectangle subdivided into squares of side 1, 1, 2, 3, 5, 8, 13… The arc inscribed across each square produces the "Fibonacci spiral," which is geometrically a quarter-circle approximation of the true logarithmic golden spiral. The two are interchangeable at sketching resolution but technically distinct — a fact discussed under golden spiral.

Etymology

Named for Leonardo of Pisa (c. 1170–c. 1240), a Tuscan mathematician who introduced the sequence to European mathematics in Liber Abaci (1202, "Book of Calculation") as the solution to a thought-experiment about rabbit population growth. "Fibonacci" — a contraction of filius Bonacci, "son of Bonacci" — was a posthumous nickname coined by the historian Guillaume Libri in 1838; Leonardo signed his own work as Leonardo Pisano. The same sequence appears earlier in Indian mathematics, in Pingala's Chandaḥśāstra (c. 200 BCE) as a count of Sanskrit metric patterns.

Examples in use

In Béla Bartók's Music for Strings, Percussion and Celesta (1936, Sz. 106), the climax of the first movement falls at bar 55 of a 89-bar structure, with internal pivots at bars 8, 13, 21, and 34. Ernő Lendvai's Béla Bartók: An Analysis of His Music (Kahn & Averill, 1971) was the first systematic study of Bartók's Fibonacci structural divisions.

In phyllotaxis — the arrangement of leaves and seeds in plants — Fibonacci numbers appear as the count of spirals visible on a sunflower head (typically 34 in one direction and 55 in the other) and the count of segments on a pinecone. Roger Jean's Phyllotaxis (Cambridge University Press, 1994) catalogues 12,000 plant samples and finds Fibonacci ratios in over 92% of them.