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what is golden ratio explain in long ...
and what it's uses. ...
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The Golden ratio is a mathematical ratio . It is a commonly found in nature, and when used in design, it fosters organic and natural looking compositions that are aesthetically pleasing to the eye..... In fact, our brains are seemingly hard- wired to prefer objects .......
mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. The figure on the right illustrates the geometric relationship. Expressed algebraically, for quantities a and b with a > b > 0,
{\displaystyle {\frac {a+b}{a}}={\frac {a}{b}}\ {\stackrel {\text{def}}{=}}\ \varphi ,} {\displaystyle {\frac {a+b}{a}}={\frac {a}{b}}\ {\stackrel {\text{def}}{=}}\ \varphi ,}
where the Greek letter phi ( {\displaystyle \varphi } \varphi or {\displaystyle \phi } \phi ) represents the golden ratio.[a] It is an irrational number that is a solution to the quadratic equation {\displaystyle x^{2}-x-1=0} x^{2}-x-1=0, with a value of:
{\displaystyle \varphi ={\frac {1+{\sqrt {5}}}{2}}=1.6180339887\ldots .} \varphi ={\frac {1+{\sqrt {5}}}{2}}=1.6180339887\ldots .[1]
The golden ratio is also called the golden mean or golden section (Latin: sectio aurea).[2][3] Other names include extreme and mean ratio,[4] medial section, divine proportion, divine section (Latin: sectio divina), golden proportion, golden cut,[5] and golden number.[6][7][8]
Mathematicians since Euclid have studied the properties of the golden ratio, including its appearance in the dimensions of a regular pentagon and in a golden rectangle, which may be cut into a square and a smaller rectangle with the same aspect ratio. The golden ratio has also been used to analyze the proportions of natural objects as well as man-made systems such as financial markets, in some cases based on dubious fits to data.[9] The golden ratio appears in some patterns in nature, including the spiral arrangement of leaves and other plant parts.