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Golden ratio

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The golden section: The golden ratio applied to the division of a line

The golden ratio, golden mean, golden number, or golden section is the mathematical constant


More importantly, it is the ratio of two quantities A and B such that the ratio from A to B (where A is the smaller one) is the same as the ratio from B to A + B; this comes from the fact that it's the positive real root of x^2-x-1 .

It has generally been thought to be pleasing and harmonious to human perception and is the basis of much classical architecture. The usage of the Greek letter phi to represent the golden ratio was suggested by mathematician Mark Barr from the first letter of Phidias (ancient Greek, Φειδίας), the sculptor who was alleged to have used it in creating statues for the Parthenon.

The golden ratio is closely associated with the Fibonacci sequence 1,1,2,3,5,8,13,21,34,\ldots Among many other things, the ratios of successive Fibonacci numbers converge to phi:

A rectangle with the ratio of adjacent sides equal to the golden ratio. It is supposedly particularly pleasant, visually.
1/1 = 1.000000
2/1 = 2.000000
3/2 = 1.500000
5/3 = 1.666666
8/5 = 1.600000
13/8 = 1.625000
21/13 = 1.615385
34/21 = 1.619048
55/34 = 1.617647
89/55 = 1.618182
144/89 = 1.617978
233/144 = 1.618056
377/233 = 1.618026
610/377 = 1.618037
987/610 = 1.618033

Phi woo[edit]

Phi and the Fibonacci numbers lend to a lot of very fascinating mathematical properties, but some cranks are willing to push it further with a good dose of pareidolia.[1][2] A classic example is nautilus shells: it is often said that they're golden spirals, when in fact they're just logarithmic spirals with ratios usually around 1.3 or so. Others claim to have found the golden ratio or Fibonacci numbers in human facial beauty, historical architecture (sometimes legitimate), Apple products, planets, musical instruments, ideal loudspeaker cabinets, ... the list goes on and on. In many cases, they've actually found the golden ratio in object X, but it's not a particularly special result; with enough perseverance, you can find the golden ratio in just about goddamn anything.

The distinction between an actual case of the golden ratio's magic and a mere crank sighting is when the presence of the golden ratio can be actually explained. That is, the golden ratio should appear in both theoretical models describing X and measured outcomes in X. Through the study of phyllotaxisWikipedia's W.svg, botanists have not only observed sunflower seeds growing in Fibonacci-numbered spirals, but they've provided a scientific explanation for why that happens: an example of the golden ratio legitimately appearing in nature.

The numerological woo surrounding phi and Fibonacci is an example of the strong law of small numbersWikipedia's W.svg; that is, there are not very many small numbers (or visually distinguishable ratios between 1 and 2) and they'll show up in many unrelated places. See also Ramsey theory.

Meanwhile, in Creationistland...


  • The Golden Ratio: The Story of Phi, The World's Most Astonishing Number by Mario Livio, 2002

See also[edit]

Mathematics Articles on RationalWiki


Conservapedian mathematics  -  Fermat's last theorem  -  Fibonacci sequence  -  Gödel's incompleteness theorems  -  Hypatia of Alexandria  -  Information  -  Mathematics  -  Metric system  -  Phli (fun)  -  Pyramid  -  Rene Descartes  -  Sophie Germain  -  Statistics  -  wikiFactor  -  Zero  -