Fibonacci sequence, the sequence of numbers 1, 1, 2, 3, 5, 8, 13, 21, …, each of which, after the second, is the sum of the two previous numbers; that is, the nth Fibonacci number Fn = Fn − 1 + Fn − 2. The sequence was noted by the medieval Italian mathematician Fibonacci (Leonardo Pisano) in his Liber abaci (1202; “Book of the Abacus”), which also popularized Hindu-Arabic numerals and the decimal number system in Europe. Fibonacci introduced the sequence in the context of the problem of how many pairs of rabbits there would be in an enclosed area if every month a pair produced a new pair and rabbit pairs could produce another pair beginning in their second month. The numbers of the sequence occur throughout nature, such as in the spirals of sunflower heads and snail shells. The ratios between successive terms of the sequence tend to the golden ratio φ = (1 + Square root of5)/2 or 1.6180…. For information on the interesting properties and uses of the Fibonacci numbers, see number games: Fibonacci numbers.

The Editors of Encyclopaedia BritannicaThis article was most recently revised and updated by Erik Gregersen.
Also known as the:
golden section, golden mean, or divine proportion

golden ratio, in mathematics, the irrational number (1 + Square root of5)/2, often denoted by the Greek letter ϕ or τ, which is approximately equal to 1.618. It is the ratio of a line segment cut into two pieces of different lengths such that the ratio of the whole segment to that of the longer segment is equal to the ratio of the longer segment to the shorter segment. The origin of this number can be traced back to Euclid, who mentions it as the “extreme and mean ratio” in the Elements. In terms of present day algebra, letting the length of the shorter segment be one unit and the length of the longer segment be x units gives rise to the equation (x + 1)/x = x/1; this may be rearranged to form the quadratic equation x2 – x – 1 = 0, for which the positive solution is x = (1 + Square root of5)/2, the golden ratio.

The ancient Greeks recognized this “dividing” or “sectioning” property, a phrase that was ultimately shortened to simply “the section.” It was more than 2,000 years later that both “ratio” and “section” were designated as “golden” by German mathematician Martin Ohm in 1835. The Greeks also had observed that the golden ratio provided the most aesthetically pleasing proportion of sides of a rectangle, a notion that was enhanced during the Renaissance by, for example, the work of the Italian polymath Leonardo da Vinci and the publication of De divina proportione (1509; Divine Proportion), written by the Italian mathematician Luca Pacioli and illustrated by Leonardo.

The golden ratio occurs in many mathematical contexts. It is geometrically constructible by straightedge and compass, and it occurs in the investigation of the Archimedean and Platonic solids. It is the limit of the ratios of consecutive terms of the Fibonacci number sequence 1, 1, 2, 3, 5, 8, 13,…, in which each term beyond the second is the sum of the previous two, and it is also the value of the most basic of continued fractions, namely 1 + 1/(1 + 1/(1 + 1/(1 +⋯.

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In modern mathematics, the golden ratio occurs in the description of fractals, figures that exhibit self-similarity and play an important role in the study of chaos and dynamical systems.

Stephan C. Carlson