Golden ratio

{{expand}}
[[Image:Golden_Ratio_Rectangle.JPG|thumb|right|200px|A rectangle with the ratio of adjacent sides equal to the golden ratio. It is supposedly particularly pleasant, visually.]]
[[Image:Golden-ratio.png|thumb|right|200px|The golden section: The golden ratio applied to the division of a line]]
{{mathematics}}
The '''golden ratio''', '''golden mean''', '''golden number''', or '''golden section''' is the mathematical constant

<math>\varphi=\frac{1+\sqrt5}{2}\approx1.6180339\ldots</math>

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 <math>x^2-x-1</math> .

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''' (φ) to represent the golden ratio was suggested by mathematician {{wpl|Mark Barr}} from the first letter of {{wpl|Phidias}} (ancient Greek, Φειδίας), the sculptor who was alleged to have used it in creating statues for the {{wpl|Parthenon||.}} 

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

{|border =0 
|-
| align="right" | 1/1 
| = 
| 1.000000
|-
| align="right" | 2/1 
| = 
| 2.000000
|-
| align="right" | 3/2 
| = 
|1.500000
|-
| align="right" | 5/3 
| = 
| 1.666666
|-
| align="right" | 8/5 
| = 
| 1.600000
|-
| align="right" | 13/8 
| = 
| 1.625000
|-
| align="right" | 21/13 
| = 
| 1.615385
|-
| align="right" | 34/21 
| = 
| 1.619048
|-
| align="right" | 55/34 
| = 
| 1.617647
|-
| align="right" | 89/55 
| = 
| 1.618182
|-
| align="right" | 144/89 
| = 
| 1.617978
|-
| align="right" | 233/144 
| = 
| 1.618056
|-
| align="right" | 377/233 
| = 
| 1.618026
|-
| align="right" | 610/377 
| = 
| 1.618037
|-
| align="right" | 987/610 
| = 
| 1.618033
|}

<!--==Derivation==-->

==Phi woo==
Phi and the Fibonacci numbers tend to have a lot of very fascinating mathematical properties, but some [[crank]]s are willing to push it further with a good dose of [[pareidolia]].<ref>[https://web.archive.org/web/20051226122417/http://www.laputanlogic.com/articles/2005/04/14-1647-4601.html The Cult of the Golden Ratio] (15 April 2005) ''Laputan Logic'' (archived from December 26, 2005).</ref><ref>[http://goldennumber.net/ Goldennumber.net], a crank site</ref> 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 Inc.|Apple products]], [[planet]]s, 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''.

[[File:Aloe polyphylla 1.jpg|thumb|right|165px| The ''Aloe polyphylla'' displaying a Fibonacci spiral-like phyllotaxis]]
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 {{wpl|phyllotaxis||,}} 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 [[numerology|numerological]] [[woo]] surrounding phi and Fibonacci is an example of the {{wpl|strong law of small numbers||;}} 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 [[Creationist]]land… [[Goddidit]].<ref>[http://creationanswers.net/inteldesign/GODRATIO.HTM God's Ratio] by Wayne Spencer, ''Creationist Answers''.</ref>

==Bibliography==
*''The Golden Ratio: The Story of Phi, The World's Most Astonishing Number'' by Mario Livio (2002) Broadway Books. ISBN 0767908155.

==References==
{{reflist}}

==See also==
*[[Pi]]

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Categories: Mathematics, Numbers