We are creating the world's most trusted encyclopedia and knowledge base.
Once you join us and log in, you'll be able to edit this page instantly!

Golden ratio

From Citizendium, the Citizens' Compendium

(Redirected from Golden section)
Jump to: navigation, search
Image:Statusbar3.png
Main Article
Talk
Definition [?]
Related Articles  [?]
Bibliography  [?]
External Links  [?]
 
This is a draft article, under development. These unapproved articles are subject to a disclaimer.

The golden ratio, also frequently known by a number of other names such as golden section or golden mean, is a mathematical proportion that is important in the arts and interesting to mathematicians. In architecture and painting, some works have been proportioned to approximate the golden ratio ever since antiquity, when, supposedly, some of the buildings on the Acropolis derived their eye-pleasing esthetics from the use of this ratio in determining the length of the buildings to their height and width.

According to the Merriam-Webster's Collegiate Dictionary, Eleventh Edition, the proportion is derived from two segments in which "the ratio of the whole to the larger part is the same as the ratio of the larger part to the smaller."

To be more elaborate: if there is a longer line segment \scriptstyle a\ and and a shorter line segment \scriptstyle b\ , and if the ratio between \scriptstyle a + b\ and \scriptstyle a\ is equal to the ratio between the line segment \scriptstyle a\ and \scriptstyle b\ , this ratio is the golden ratio. The value of the golden ratio is \scriptstyle \Phi = \frac{a}{b}= \frac{1 + \sqrt{5}}{2} = 1{,}618033988\dots

Properties

  • If \ \scriptstyle \frac{a}{b}= \frac{1 + \sqrt{5}}{2}  it follows that \ \scriptstyle \frac{a}{b}= \frac{a+b}{a} = 1 + \frac{b}{a}

With \scriptstyle \Phi = 1 + \frac{1}{\Phi} we could derive the infinite continued fraction of the golden ratio: \Phi = 1 + \frac{1}{\Phi} = 1 + \frac{1}{1 + \frac{1}{\Phi}} = 1 + \frac{1}{1 + \frac{1}{ 1 + \frac{1}{\Phi}}} = \dots

Thus

\Phi\ =\ 1 + \frac{1}{1 + \frac{1}{ 1 + \frac{1}{1 + \dots}}}


  • \Phi\ =\ \lim_{n\to\infty}\frac{F(n+1)}{F(n)},

where \ F_n is the n-th term of the Fibonacci sequence.

  • The golden ratio is irrational and, in a sense, the hardest among irrational numbers to approximate by rational numbers. Only rational numbers are harder to approximate by other rational numbers. Thus one may say that of all irrational numbers the golden ratio is the least irrational.
Retrieved from "http://en.citizendium.org/wiki/Golden_ratio"
Views
Personal tools