Antiparallel (mathematics): Difference between revisions
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{{Portal:Mathematics/Feature article|img=Pentagram-phi.svg|img-cap=A [[pentagram]] colored to distinguish its line segments of different lengths. The four lengths are in golden ratio to one another|img-cred=[[User:PAR]]|more=Golden ratio|desc=In [[mathematics]] and the [[art]]s, two quantities are in the '''golden ratio''' if the [[ratio]] between the sum of those quantities and the larger one is the same as the ratio between the larger one and the smaller. The [[golden ratio]] is a [[mathematical constant]], usually denoted by the [[Greek alphabet|Greek]] letter ''φ'' ([[phi]]). | |||
Expressed algebraically, two quantities ''a'' and ''b'' are therefore in the golden ratio if | |||
:<math> \frac{a+b}{a} = \frac{a}{b} = \varphi\,.</math> | |||
It follows from this property that ''φ'' satisfies the [[quadratic equation]] ''φ''<sup>2</sup> = ''φ'' + 1 and is therefore an [[algebraic number|algebraic]] [[irrational number]], given by | |||
:<math>\varphi = \frac{1 + \sqrt{5}}{2},\,</math> | |||
which is approximately equal to 1.6180339887. | |||
At least since the [[Renaissance]], many [[artist]]s and [[architect]]s have proportioned their works to approximate the golden ratio—especially in the form of the '''[[golden rectangle]]''', in which the ratio of the longer side to the shorter is the golden ratio—believing this proportion to be [[aesthetics|aesthetically]] pleasing. [[Mathematician]]s have studied the golden ratio because of its unique and interesting properties. | |||
Other names frequently used for or closely related to the golden ratio are '''golden section''' (Latin: ''sectio aurea''), '''golden mean''', '''golden number''', '''divine proportion''' (Italian: ''proporzionedivina''), '''divine section''' (Latin: ''sectio divina''), '''golden proportion''', '''golden cut''', and '''mean of [[Phidias]]'''.|class={{{class}}}}} | |||
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Revision as of 03:39, 15 June 2013
{{../Header}} Template:Portal:Mathematics/Feature article {{../Footer}}