Golden ratio

2017-08-06T16:18:50Z

2606:A000:4851:9D00:2D1C:CDAE:2EA2:61F1: /* Golden ratio in nature */ Someone wrote an unrelated comment ("____ was here") that doesn't contribute to the article.

If a person has one number ''a'' and another smaller number ''b'', he can make the [[ratio]] of the two numbers by [[division (mathematics)|dividing]] them. Their ratio is ''a''/''b''. The person can make another ratio by adding the two numbers together ''a''+''b'' and dividing this by the larger number ''a''. The new ratio is (''a''+''b'')/''a''. If these two ratios are equal to the same number, then that number is called the '''golden ratio'''. The [[Greek alphabet|Greek]] letter <math>\varphi</math> ([[Phi (letter)|phi]]) is usually used as the name for the golden ratio.

For example, if ''b'' = 1 and ''a''/''b'' = <math>\varphi</math>, then ''a'' = <math>\varphi</math>. The second ratio (''a''+''b'')/''a'' is then <math>(\varphi+1)/\varphi</math>. Because these two ratios are equal, this is true:
<blockquote>
<math>\varphi = \frac{\varphi+1}{\varphi}</math>
</blockquote>
One way to write this number is<blockquote>
<math>\varphi = \frac{1 + \sqrt{5}}{2}</math>
</blockquote>
<math>\sqrt{5}</math> is like any number which, when multiplied by itself, makes 5(or which number is multiplied): <math>\sqrt5\times\sqrt5=5</math>.

The golden ratio is an [[Number#Irrational numbers|irrational number]]. If a person tries to write it, it will never stop and never be the same again and again, but it will start this way: 1.6180339887... An important thing about this number is that a person can subtract 1 from it or divide 1 by it. Either way, the number will still keep going and never stop.
:<math>\begin{array}{ccccc}
\varphi-1 &=& 1.6180339887...-1 &=& 0.6180339887...\\
1/\varphi &=& \frac{1}{1.6180339887...} &=& 0.6180339887...
\end{array}</math>

== Golden rectangle ==

[[Image:Golden-rectangle-detailed.svg|thumb|240px|The large rectangle '''BA''' is a golden rectangle; that is, the proportion b:a is 1:<math>\varphi</math>. For any such rectangle, and only for rectangles of that specific proportion, if we remove square '''B''', what is left, '''A''', is another golden rectangle; that is, with the same proportions as the original rectangle.]]

If the length of a [[rectangle]] divided by its width is equal to the golden ratio, then the rectangle is a "golden rectangle". If a square is cut off from one end of a golden rectangle, then the other end is a new golden rectangle. In the picture, the big rectangle (blue and pink together) is a golden rectangle because <math>a/b=\varphi</math>. The blue part (B) is a square. The pink part by itself (A) is another golden rectangle because <math>b/(a-b)=\varphi</math>. The big rectangle and the pink rectangle have the same form, but the pink rectangle is smaller and is turned.

== Fibonacci numbers ==

The [[Fibonacci]] numbers are a list of numbers. A person can find the next number in the list by adding the last two numbers together. If a person divides a number in the list by the number that came before it, this ratio comes closer and closer to the golden ratio.
{| class="wikitable"
|-
! Fibonacci number
! divided by the one before
! ratio
|-
| 1
|
|
|-
| 1
| 1/1
| = 1.0000
|-
| 2
| 2/1
| = 2.0000
|-
| 3
| 3/2
| = 1.5000
|-
| 5
| 5/3
| = 1.6667
|-
| 8
| 8/5
| = 1.6000
|-
| 13
| 13/8
| = 1.6250
|-
| 21
| 21/13
| = 1.6154...
|-
| 34
| 34/21
| = 1.6190...
|-
| 55
| 55/34
| = 1.6177...
|-
| 89
| 89/55
| = 1.6182...
|-
| ...
| ...
| ...
|-
|
| <math>\varphi</math>
| = 1.6180...
|}

==Golden ratio in nature==
[[File:Goldener Schnitt Blattstand.png|thumb|Using the golden angle will optimally use the light of the sun. This is a view from the top.]]
[[File:Efeublatt.jpg|thumb|A leaf of [[common ivy]], showing the golden ratio]]
In [[nature]], the golden ratio is often used for the arrangement of [[leaf|leaves]] or [[flower]]s. These use the golden angle of approximately 137.5 degrees. Leaves or flowers arranged in that angle best use sunlight.

==Other websites==
*[http://www.mathsisfun.com/numbers/golden-ratio.html Easy to understand golden ratio article]

[[Category:Irrational numbers]]

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Golden ratio