Golden ratio: Difference between revisions
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If there is a longer line segment <math>\scriptstyle a\ </math> and and a shorter line segment <math>\scriptstyle b\ </math>, and if the ratio between <math>\scriptstyle a + b\ </math> and <math>\scriptstyle a\ </math> is equal to the ratio between the line segment <math>\scriptstyle a\ </math> and <math>\scriptstyle b\ </math>, this ratio is called '''golden ratio'''. The value of the golden ratio is <math>\scriptstyle \Phi = \frac{a}{b}= \frac{1 + \sqrt{5}}{2} = 1{,}618033988\dots</math> | If there is a longer line segment <math>\scriptstyle a\ </math> and and a shorter line segment <math>\scriptstyle b\ </math>, and if the ratio between <math>\scriptstyle a + b\ </math> and <math>\scriptstyle a\ </math> is equal to the ratio between the line segment <math>\scriptstyle a\ </math> and <math>\scriptstyle b\ </math>, this ratio is called '''golden ratio'''. The value of the golden ratio is <math>\scriptstyle \Phi = \frac{a}{b}= \frac{1 + \sqrt{5}}{2} = 1{,}618033988\dots</math> | ||
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With <math>\scriptstyle \Phi = 1 + \frac{1}{\Phi}</math> we could derive the infinate [[continued fraction]] of the golden ratio: | With <math>\scriptstyle \Phi = 1 + \frac{1}{\Phi}</math> we could derive the infinate [[continued fraction]] of the golden ratio: | ||
<math>\Phi = 1 + \frac{1}{\Phi} = 1 + \frac{1}{1 + \frac{1}{\Phi}} = 1 + \frac{1}{1 + \frac{1}{ 1 + \frac{1}{\Phi}}} = \dots = 1 + \frac{1}{1 + \frac{1}{ 1 + \frac{1}{1 + \dots}}}</math> | <math>\Phi = 1 + \frac{1}{\Phi} = 1 + \frac{1}{1 + \frac{1}{\Phi}} = 1 + \frac{1}{1 + \frac{1}{ 1 + \frac{1}{\Phi}}} = \dots = 1 + \frac{1}{1 + \frac{1}{ 1 + \frac{1}{1 + \dots}}}</math> | ||
Revision as of 01:16, 23 December 2007
If there is a longer line segment and and a shorter line segment , and if the ratio between and is equal to the ratio between the line segment and , this ratio is called golden ratio. The value of the golden ratio is
Properties
If it follows that
With we could derive the infinate continued fraction of the golden ratio:
